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Download Doc 126 B.P.S. XII Physics IIT JEE Advanced Study Package 2014 15...
BRILLIANT PUBLIC SCHOOL, SITAMARHI
(Affiliated up to +2 level to C.B.S.E., New Delhi)
Class-XII IIT-JEE Advanced Physics Study Package Session: 2014-15 Office: Rajopatti, Dumra Road, Sitamarhi (Bihar), Pin-843301 Ph.06226-252314 , Mobile:9431636758, 9931610902 Website: www.brilliantpublicschool.com; E-mail:
[email protected]
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: PHYSICS-XII Chapters: 1. Electrostatics 2. Capacitance 3. Current Electricity 4. Thermal and Chemical Effects of Electric Current 5. Magnetic Effects of Electric Current 6. Electromagnetic Induction and Alternating Current 7. Optics 8. Optical Instruments 9. Wave Optics 10. Modern Physics 11. Semiconductors Electronics
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: PHYSICS TOPIC: XII P1. Electrostatics Index: 1. Key Concepts 2. Exercise I 3. Exercise II 4. Exercise III 5. Exercise IV 6. Answer Key 7. 34 Yrs. Que. from IIT-JEE 8. 10 Yrs. Que. from AIEEE
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1.
2.
ELECTRIC CHARGE Charge of a material body is that possesion (acquired or natural) due to which it strongly interacts with other material body. It can be postive or negative. S.I. unit is coulomb. Charge is quantized, conserved, and additive. 1 q1q 2 1 q1q 2 F = r where . In vector form 4πε 0ε r r 3 4πε 0ε r r 2 ε0 = permittivity of free space = 8.85 × 10−12 N−1 m−2 c2 or F/m and εr = Relative permittivity of the medium = Spec. Inductive Capacity = Dielectric Const. ε0εr = Absolute permittivity of the medium εr = 1 for air (vacuum) = ∞ for metals
COULOMB’S LAW : F =
NOTE : The Law is applicable only for static and point charges. Only applicable to static charges as moving charges may result magnetic interaction also and only for point charges as if charges are extended, induction may change the charge distribution. 3.
PRINCIPLE OF SUPER POSITION Force on a point charge due to many charges is given by F=F1+F2 +F3 +..........
NOTE : The force due to one charge is not affected by the presence of other charges. 4.
ELECTRIC FIELD, ELECTRIC INTENSITY OR ELECTRIC FIELD STRENGTH (VECTOR QUANTITY) “The physical field where a charged particle, irrespective of the fact whether it is in motion or at rest, experiences force is called an electric field”. The direction of the field is the direction of the force experienced by a positively charged particle & the magnitude of the field (electric intensity) is F Lim unit is NC–1 ; S.I. unit is the force experienced by the particle carrying unit charge E = q → 0 q V/m here Lim represents that this charge does not alter the magnitude of electric field. Due to q →0 charge induction on the source of electric field.
(i)
ELECTRIC FIELD DUE TO 1 q 1 q r (vector form) rˆ = Point charge : E = 2 4π ∈0 r 3 4πε 0 r
(ii)
Where r = vector drawn from the source charge to the point . 1 dq ˆ Continuous charge distribution E = r = d E ; dE = electric field due to an elementry charge 4πε 0 ∫ r 2 ∫
5.
. Note E ≠ ∫ dE because E is a vector quantity .
(iii)
dq = λ dl (for line charge) = σ ds (for surface charge) = ρ dv (for volume charge) In general λ, σ & ρ are linear, surface and volume charge densities respectively. 2k λ Infinite line of charge E = where r = perpendicular distance of the point from the line charge . r
(iv)
Semi ∞ line of charge E =
kλ kλ 2 kλ as , Ex = & Ey = at a point above the end of wire at r r r
an angle 45º .
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KEY CONCEPTS
Uniformly charged ring , Ecentre = 0 , Eaxis =
(vi)
Electric field is maximum when
(vii)
kQx ( x + R 2 )3 / 2 2
dE = 0 for a point on the axis of the ring. Here we get x = R/√2. dx σ Infinite non conducting sheet of charge E = nˆ where 2ε 0
= unit normal vector to the plane of sheet, where σ is surface charge density ∞ charged conductor sheet having surface charge density σ on both surfaces E = σ/ε0 . n
(viii) (ix) (x)
Just outside a conducting surface charged with a surface charge density σ, electric field is always given as E = σ/∈0. Q Uniformly charged solid sphere (Insulating material) E out = ; r ≥R, 4πε 0 r 2 Behaves as a point charge situated at the centre for these points Ein =
Qr ρr = ; 3 4πε 0 R 3ε 0
r ≤ R where ρ = volume charge density (xi)
Uniformly charged spherical shell (conducting or non-donducting) or uniformly charged solid Q ; r ≥ R conducting sphere . Eout = 4πε 0 r 2 Behaves as a point charge situated at the centre for these points E in = 0 ; r < R
(xii)
(xiii)
6.
(i) (ii) (iii) (iv) (v) 7. (i) (ii)
uniformly charged cylinder with a charge density ρ is -(radius of cylinder = R) for r < R ρr ρR 2 Em = 2 ∈ ; for r > R E= 2 ∈0 r 0 Uniformly charged cylinderical shell with surface charge density σ is ρr for r < R Em = 0 ; for r > R E = ∈ r 0 ELECTRIC LINES OF FORCE (ELF) The line of force in an electric field is a hypothetical line, tangent to which at any point on it represents the direction of electric field at the given point. Properties of (ELF) : Electric lines of forces never intersects . ELF originates from positive charge or ∞ and terminate on a negative charge of infinity . Preference of termination is towards a negative charge . If an ELF is originated, it must require termination either at a negetive charge or at ∞ . Quantity of ELF originated or terminated from a charge or on a charge is proportional to the magnitude of charge. ELECTROSTATIC EQUILIBRIUM Position where net force (or net torque) on a charge(or electric dipole) = 0 STABLE EQUILIBRIUM : If charge is displaced by a small distance the charge comes (or tries to come back) to the equilibrium . UNSTABLE EQUILIBRIUM : If charge is displaced by a small distance the charge does not return to the equilibrium position.
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(v)
ELECTRIC POTENTIAL (Scalar Quantity) “Work done by external agent to bring a unit positive charge(without accelaration) from infinity to a point in an electric field is called electric potential at that point” . If W∞ r is the work done to bring a charge q (very small) from infinity to a point then potential at that
point is V =
9.
( W∞r ) ext
; S.I. unit is volt ( = 1 J/C) q POTENTIAL DIFFERENCE VAB = VA − VB =
( WBA ) ext
VAB = p.d. between point A & B . q WBA = w.d. by external source to transfer a point charge q from B to A (Without acceleration).
10.
ELECTRIC FIELD & ELECTRIC POINTENIAL ˆ∂ ˆ∂ ˆ∂ E = − grad V = − ∇ V {read as gradient of V} grad = i + j +k
∂x
∂y
∂z
;
Used when EF varies in three dimensional coordinate system. For finding potential difference between two points in electric field, we use – B →→ − VA – VB = ∫ E .dt if E is varying with distance A
if E is constant & here d is the distance between points A and B.
(i)
POTENTIAL DUE TO Q a point charge V = 4πε 0 r
(iii)
continuous charge distribution V =
(iv)
spherical shell (conducting or non conducting) or solid conducting sphere Q Q Vout = ; (r ≥ R) , Vin = ; (r ≤ R) 4 πε 0 r 4πε0 R non conducting uniformly charged solid sphere :
11.
(v)
Vout = 12.
(ii)
Q ; (r ≥ R) , 4πε 0 r
many charges V =
q1
+
q2
+
q3
4πε 0 r1 4 πε 0 r2 4 πε 0 r3
+ ......
1 dq ∫ 4πε 0 r
Vin =
1 Q(3R 2 −r 2 ) ; (r ≤ R) 2 4πε 0 R
EQUIPOTENTIAL SURFACE AND EQUIPOTENTIAL REGION In an electricfield the locus of points of equal potential is called an equipotential surface. An equipotential surface and the electric field meet at right angles.
The region where E = 0, Potential of the whole region must remain constant as no work is done in displacement of charge in it. It is called as equipotential region like conducting bodies.
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8.
MUTUAL POTENTIAL ENERGY OR INTERACTION ENERGY “The work to be done to integrate the charge system .” qq For 2 particle system Umutual = 1 2 4πε 0 r
For 3 particle system Umutual = For n particles there will be
q q q q q1q 2 + 2 3 + 3 1 4πε 0 r12 4πε 0 r23 4πε 0 r31
n (n −1) terms . Total energy of a system = Uself + Umutual 2
14.
P.E. of charge q in potential field U = qV. Interaction energy of a system of two charges U = q 1 V 2 = q 2 V1 .
15. (a)
ELECTRIC DIPOLE. O is mid point of line AB (centre of the dipole) on the axis (except points on line AB) pr p E= ≈ 2πε [r 2 − (a 2 / 4)]2 2πε r 3 ( if r < < a) 0
p = q a = Dipole moment ,
(b) (c)
(d)
0
r = distance of the point from the centre of dipole p p ≈ − on the equitorial ; E = 4πε 0 [ r 2 + (a 2 / 4)]3 / 2 4πε 0 r 3 At a general point P(r, θ) in polar co-ordinate system is 2kp sin θ Radial electric field Er = r3 kp cosθ Tangentral electric field ET = r3 2 2 kp 2 Net electric field at P is Enet = E r + E T = 3 1 + 3 sin θ r kp sin θ Potential at point P is VP = r2 NOTE : If θ is measured from axis of dipole. Then sinθ and cosθ will be interchanged. Pθ p.r Dipole V = = p=qa electric dipole moment . If θ is angle between p and 2 3 4πε 0 r 4πε 0 r reaches vector of the point.
(e) (f) (g)
Electric Dipole in uniform electric field : torque τ=px E ; F = 0 . Work done in rotation of dipole is w = PE (cos θ1 − cos θ2) P.E. of an electric dipole in electric field U = − p.E . dE d ˆ Force on a dipole when placed in a non uniform electric field is F=− − P.E i = P. ˆi . dx dx
(
)
16. (i)
ELECTRIC FLUX For uniform electric field; φ = E . A = EA cos θ where θ = angle between E & area vector ( A ). Flux is contributed only due to the component of electric field which is perpendicular to the plane.
(ii)
If E is not uniform throughout the area A , then φ = ∫ E.dA
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13.
GAUSS’S LAW (Applicable only to closed surface) “ Net flux emerging out of a closed surface is q q .” φ = ∫ EdA = q = net charge enclosed by the closed surface . ε0 ε0
φ does not depend on the
18. 19. 20.
(i) (ii)
shape and size of the closed surface The charges located outside the closed surface.
CONCEPT OF SOLID ANGLE : Flux of charge q having through the circle of radius R is q q / ∈0 x Ω = 2 ∈ (1 – cosθ) φ= 4π 0 ε E2 Energy stored p.u. volume in an electric field = 0 2 σ2 Electric pressure due to its own charge on a surface having charged density σ is Pele = . 2ε 0 Electric pressure on a charged surface with charged density σ due to external electric field is Pele = σE1 IMPORTANT POINTS TO BE REMEMBERED
(i)
Electric field is always perpendicular to a conducting surface (or any equipotential surface) . No tangential component on such surfaces .
(ii)
Charge density at sharp points on a conductor is greater.
(iii)
When a conductor is charged, the charge resides only on the surface.
(iv)
For a conductor of any shape E (just outside) =
(v)
p.d. between two points in an electric field does not depend on the path joining them .
(vi)
Potential at a point due to positive charge is positive & due to negative charge is negative.
(vii)
Positive charge flows from higher to lower (i.e. in the direction of electric field) and negative charge from lower to higher (i.e. opposite to the electric field) potential . When p||E the dipole is in stable equilibrium p||( − E ) the dipole is in unstable equilibrium
(viii) (ix)
σ ε0
(x)
When a charged isolated conducting sphere is connected to an unchaged small conducting sphere then potential (and charge) remains almost same on the larger sphere while smaller is charged .
(xi)
Self potential energy of a charged shell =
(xii) (xiii) (xiv)
KQ 2 . 2R
3k Q 2 . 5R A spherically symmetric charge {i.e ρ depends only on r} behaves as if its charge is concentrated at its centre (for outside points).
Self potential energy of an insulating uniformly charged sphere =
Dielectric strength of material : The minimum electric field required to ionise the medium or the maximum electric field which the medium can bear without breaking down.
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17.
Q.1
A negative point charge 2q and a positive charge q are fixed at a distance l apart. Where should a positive test charge Q be placed on the line connecting the charge for it to be in equilibrium? What is the nature of the equilibrium with respect to longitudinal motions?
Q.2
(a)
Two particles A and B each carrying a charge Q are held fixed with a separation d between then A particle C having mass m ans charge q is kept at the midpoint of line AB. If it is displaced through a small distance x (x > a) Q.6
The figure shows three infinite non-conducting plates of charge perpendicular to the plane of the paper with charge per unit area + σ, + 2σ and – σ. Find the ratio of the net electric field at that point A to that at point B.
Q.7
A thin circular wire of radius r has a charge Q. If a point charge q is placed at the centre of the ring, then find the increase in tension in the wire.
Q.8
In the figure shown S is a large nonconducting sheet of uniform charge density σ. A rod R of length l and mass ‘m’ is parallel to the sheet and hinged at its mid point. The linear charge densities on the upper and lower half of the rod are shown in the figure. Find the angular acceleration of the rod just after it is released.
Q.9
A simple pendulum of length l and bob mass m is hanging in front of a large nonconducting sheet having surface charge density σ. If suddenly a charge +q is given to the bob & it is released from the position shown in figure. Find the maximum angle through which the string is deflected from vertical.
Q.10 A particle of mass m and charge – q moves along a diameter of a uniformly charged sphere of radius R and carrying a total charge + Q. Find the frequency of S.H.M. of the particle if the amplitude does not exceed R. Q.11
A charge + Q is uniformly distributed over a thin ring with radius R. A negative point charge – Q and mass m starts from rest at a point far away from the centre of the ring and moves towards the centre. Find the velocity of this particle at the moment it passes through the centre of the ring.
Q.12 A spherical balloon of radius R charged uniformly on its surface with surface density σ. Find work done against electric forces in expanding it upto radius 2R.
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EXERCISE # I
Q.14 Consider the configuration of a system of four charges each of value +q. Find the work done by external agent in changing the configuration of the system from figure (i) to fig (ii).
Q.15 There are 27 drops of a conducting fluid. Each has radius r and they are charged to a potential V0. They are then combined to form a bigger drop. Find its potential. Q.16 Two identical particles of mass m carry charge Q each. Initially one is at rest on a smooth horizontal plane and the other is projected along the plane directly towards the first from a large distance with an initial speed V. Find the closest distance of approach. Q.17 A particle of mass m and negative charge q is thrown in a gravity free space with speed u from the point A on the large non conducting charged sheet with surface charge density σ, as shown in figure. Find the maximum distance from A on sheet where the particle can strike. Q.18 Consider two concentric conducting spheres of radii a & b (b > a). Inside sphere has a positive charge q1. What charge should be given to the outer sphere so that potential of the inner sphere becomes zero? How does the potential varies between the two spheres & outside ? Q.19 Three charges 0.1 coulomb each are placed on the corners of an equilateral triangle of side 1 m. If the energy is supplied to this system at the rate of 1 kW, how much time would be required to move one of the charges onto the midpoint of the line joining the other two? Q.20 Two thin conducting shells of radii R and 3R are shown in figure. The outer shell carries a charge +Q and the inner shell is neutral. The inner shell is earthed with the help of switch S. Find the charge attained by the inner shell. Q.21 Consider three identical metal spheres A, B and C. Spheres A carries charge + 6q and sphere B carries charge – 3q. Sphere C carries no charge. Spheres A and B are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Finally the sphere C is touched to sphere B and separated from it. Find the final charge on the sphere C. Q.22 A dipole is placed at origin of coordinate system as shown in figure, find the electric field at point P (0, y). pˆ k are located at (0, 0, 0) and (1m, 0, 2m) respectively. Find the resultant 2 electric field due to the two dipoles at the point (1m, 0, 0).
Q.23 Two point dipoles p kˆ and
Q.24 The length of each side of a cubical closed surface is l. If charge q is situated on one of the vertices of the cube, then find the flux passing through shaded face of the cube. Q.25 A point charge Q is located on the axis of a disc of radius R at a distance a from the plane of the disc. If one fourth (1/4th) of the flux from the charge passes through the disc, then find the relation between a & R. Q.26 A charge Q is uniformly distributed over a rod of length l. Consider a hypothetical cube of edge l with the centre of the cube at one end of the rod. Find the minimum possible flux of the electric field through the entire surface of the cube.
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Q.13 A point charge + q & mass 100 gm experiences a force of 100 N at a point at a distance 20 cm from a long infinite uniformly charged wire. If it is released find its speed when it is at a distance 40 cm from wire
Q.1
(a) (c)
Q.2
A rigid insulated wire frame in the form of a right angled triangle ABC, is set in a vertical plane as shown. Two bead of equal masses m each and carrying charges q1 & q2 are connected by a cord of length 1 & slide without friction on the wires. Considering the case when the beads are stationary, determine. The angle α. (b) The tension in the cord & The normal reaction on the beads. If the cord is now cut, what are the values of the charges for which the beads continue to remain stationary.
ke 2 each, when ml they are far away from each other, as shown. The distance between their initial velocities is L. Find their closest approach distance, mass of proton=m, charge=+e, mass of α-particle = 4m, charge = + 2e. A proton and an α-particle are projected with velocity v0 =
Q.3
A clock face has negative charges − q, − 2q, − 3q, ........., − 12q fixed at the position of the corresponding numerals on the dial. The clock hands do not disturb the net field due to point charges. At what time does the hour hand point in the same direction is electric field at the centre of the dial.
Q.4
A circular ring of radius R with uniform positive charge density λ per unit length is fixed in the Y−Z plane with its centre at the origin O. A particle of mass m and positive charge q is projected from the point P
(
)
3 R,0,0 on the positive X-axis directly towards O, with initial velocity v . Find the smallest value of the speed v such that the particle does not return to P.
Q.5
2 small balls having the same mass & charge & located on the same vertical at heights h1 & h2 are thrown in the same direction along the horizontal at the same velocity v . The 1st ball touches the ground at a distance l from the initial vertical . At what height will the 2nd ball be at this instant ? The air drag & the charges induced should be neglected.
Q.6
Two concentric rings of radii r and 2r are placed with centre at origin. Two charges +q each are fixed at the diametrically opposite points of the rings as shown in figure. Smaller ring is now rotated by an angle 90° about Z-axis then it is again rotated by 90° about Y-axis. Find the work done by electrostatic forces in each step. If finally larger ring is rotated by 90° about X-axis, find the total work required to perform all three steps.
Q.7
A positive charge Q is uniformly distributed throughout the volume of a dielectric sphere of radius R . A point mass having charge + q and mass m is fired towards the centre of the sphere with velocity v from a point at distance r (r > R) from the centre of the sphere. Find the minimum velocity v so that it can penetrate R/2 distance of the sphere. Neglect any resistance other than electric interaction. Charge on the small mass remains constant throughout the motion.
Q.8
An electrometer consists of vertical metal bar at the top of which is attached a thin rod which gets deflected from the bar under the action of an electric charge (fig.) . The reading are taken on a quadrant graduated in degrees . The length of the rod is l and its mass is m . What will be the charge when the rod of such an electrometer is deflected through an angle α . Make the following assumptions : the charge on the electrometer is equally distributed between the bar & the rod the charges are concentrated at point A on the rod & at point B on the bar.
(a) (b)
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EXERCISE # II
A cavity of radius r is present inside a solid dielectric sphere of radius R, having a volume charge density of ρ. The distance between the centres of the sphere and the cavity is a . An electron e is kept inside the cavity at an angle θ = 45° as shown . How long will it take to touch the sphere again?
Q.10 Two identical balls of charges q1 & q2 initially have equal velocity of the same magnitude and direction. After a uniform electric field is applied for some time, the direction of the velocity of the first ball changes by 60º and the magnitude is reduced by half . The direction of the velocity of the second ball changes there by 90º. In what proportion will the velocity of the second ball changes ? Q.11
Electrically charged drops of mercury fall from altitude h into a spherical metal vessel of radius R in the upper part of which there is a small opening. The mass of each drop is m & charge is Q. What is the number 'n' of last drop that can still enter the sphere. Given that the (n + 1)th drop just fails to enter the sphere.
Q.12 Small identical balls with equal charges are fixed at vertices of regular 2004 - gon with side a. At a certain instant, one of the balls is released & a sufficiently long time interval later, the ball adjacent to the first released ball is freed. The kinetic energies of the released balls are found to differ by K at a sufficiently long distance from the polygon. Determine the charge q of each part. E x Q.13 The electric field in a region is given by E = 0 i . Find the charge contained inside a cubical volume l bounded by the surfaces x = 0, x = a, y = 0, y = a, z = 0 and z = a. Take E0 = 5 × 103N/C, l = 2cm and a = 1cm.
Q.14 2 small metallic balls of radii R1 & R2 are kept in vacuum at a large distance compared to the radii. Find the ratio between the charges on the 2 balls at which electrostatic energy of the system is minimum. What is the potential difference between the 2 balls? Total charge of balls is constant. Q.15 Figure shows a section through two long thin concentric cylinders of radii a & b with a < b . The cylinders have equal and opposite charges per unit length λ . Find the electric field at a distance r from the axis for (a) r < a (b) a < r < b (c) r > b Q.16 A solid non conducting sphere of radius R has a non-uniform charge distribution of volume charge r density, ρ = ρ0 , where ρ0 is a constant and r is the distance from the centre of the sphere. Show that: R (a) the total charge on the sphere is Q = π ρ0 R3 and 2
(b)
the electric field inside the sphere has a magnitude given by, E = KQr . R4
Q.17 A nonconducting ring of mass m and radius R is charged as shown. The charged density i.e. charge per unit length is λ. It is then placed on a rough nonconducting horizontal surface plane. At time t = 0, a uniform electric field E = E 0i is switched on and the ring start rolling without sliding. Determine the friction force (magnitude and direction) acting on the ring, when it starts moving.
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Q.9
Q.19 An electron beam after being accelerated from rest through a potential difference of 500 V in vacuum is allowed to impinge normally on a fixed surface. If the incident current is 100 µ A, determine the force exerted on the surface assuming that it brings the electrons to rest. (e = 1.6×10−19 C ; m = 9.0×10−31 kg)
Q.20 Find the electric field at centre of semicircular ring shown in figure.
Q.21 A cone made of insulating material has a total charge Q spread uniformly over its sloping surface. Calculate the energy required to take a test charge q from infinity to apex A of cone. The slant length is L. Q.22 An infinite dielectric sheet having charge density σ has a hole of radius R in it. An electron is released on the axis of the hole at a distance 3R from the centre. What will be the velocity which it crosses the plane of sheet. (e = charge on electron and m = mass of electron)
Q.23 Two concentric rings, one of radius 'a' and the other of radius 'b' have the charges +q and – (2 5)−3 / 2 q respectively as shown in the figure. Find the ratio b/a if a charge particle placed on the axis at z = a is in equilibrium. Q.24 Two charges + q1 & − q2 are placed at A and B respectively. A line of force emerges from q1 at angle α with line AB. At what angle will it terminate at − q2?
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Q.18 Two spherical bobs of same mass & radius having equal charges are suspended from the same point by strings of same length. The bobs are immersed in a liquid of relative permittivity εr & density ρ0. Find the density σ of the bob for which the angle of divergence of the strings to be the same in the air & in the liquid ?
EXERCISE # III
Q.1
The magnitude of electric field in the annular region of charged cylindrical capacitor (A) Is same throughout (B) Is higher near the outer cylinder than near the inner cylinder (C) Varies as (1/r) where r is the distance from the axis (D) Varies as (1/r2) where r is the distance from the axis [IIT '96, 2]
Q.2
A metallic solid sphere is placed in a uniform electric field. The lines of force follow the path (s) shown in figure as : (A) 1 (B) 2 (C) 3 (D) 4 [IIT'96 , 2]
Q.3
A non-conducting ring of radius 0.5 m carries a total charge of 1.11 × 10−10 C distributed non-uniformly on its circumference producing an electric field E every where in space. The value of the line integral =0
∫ =∞
−E.d (l = 0 being centre of the ring) in volts is :
(A) + 2 Q.4 (i)
(ii)
(iii)
Q.5
(a) (b)
(B) − 1
(C) − 2
(D) zero[JEE '97, 1 ]
Select the correct alternative : [JEE '98 2 + 2 + 2 = 6 ] A + ly charged thin metal ring of radius R is fixed in the xy−plane with its centre at the origin O . A – ly charged particle P is released from rest at the point (0, 0, z0) where z0 > 0 . Then the motion of P is: (A) periodic, for all values of z0 satisfying 0 < z0 < ∞ (B) simple harmonic, for all values of z0 satisfying 0 < z0 ≤ R (C) approximately simple harmonic, provided z0 R, mesured from the axis
Q.20 A square current carrying loop made of thin wire and having a mass m =10g can rotate without friction with respect to the vertical axis OO1, passing through the centre of the loop at right angles to two opposite sides of the loop. The loop is placed in a homogeneous magnetic field with an induction B = 10-1 T directed at right angles to the plane of the drawing. A current I = 2A is flowing in the loop. Find the period of small oscillations that the loop performs about its position of stable equilibrium.
7
Q.21 A charged particle having mass m and charge q is accelerated by a potential difference V, it flies through a uniform transverse magnetic field B. The field occupies a region of space d. Find the time interval for which it remains inside the magnetic field. Q.22 A proton beam passes without deviation through a region of space where there are uniform transverse mutually perpendicular electric and magnetic field with E and B. Then the beam strikes a grounded target. Find the force imparted by the beam on the target if the beam current is equal to I. Q.23 An infinitely long straight wire carries a conventional current I as shown in the figure. The rectangular loop carries a conventional current I' in the clockwise direction. Find the net force on the rectangular loop.
Q.24 An arc of a circular loop of radius R is kept in the horizontal plane and a constant magnetic field B is applied in the vertical direction as shown in the figure. If the arc carries current I then find the force on the arc. Q.25 Two long straight parallel conductors are separated by a distance of r1 = 5cm and carry currents i1 = 10 A & i2 = 20 A . What work per unit length of a conductor must be done to increase the separation between the conductors to r2 = 10 cm if , currents flow in the same direction? List of recommended questions from I.E. Irodov. 3.220, 3.223, 3.224, 3.225, 3.226, 3.227, 3.228, 3.229, 3.230, 3.234, 3.236, 3.237, 3.242 3.243, 3.244, 3.245, 3.251, 3.252, 3.253, 3.254, 3.257, 3.258, 3.269, 3.372, 3.373, 3.383, 3.384, 3.386, 3.389, 3.390, 3.391, 3.396
8
EXERCISE # II Q.1
Three infinitely long conductors R, S and T are lying in a horizontal plane as shown in the figure. The currents in the respective conductors are IR = I0sin (ωt +
2π ) 3
IS = I0sin (ωt) 2π ) 3 Find the amplitude of the vertical component of the magnetic field at a point P, distance 'a' away from the central conductor S.
IT = I0sin (ωt −
Q.2
Four long wires each carrying current I as shown in the figure are placed at the points A, B, C and D. Find the magnitude and direction of (i) magnetic field at the centre of the square. (ii) force per metre acting on wire at point D.
Q.3
An infinite wire, placed along z-axis, has current I1 in positive z-direction. Aconducting rod placed in xy plane parallel to y-axis has current I2 in positive y-direction. The ends of the rod subtend + 30° and – 60° at the origin with positive x-direction. The rod is at a distance a from the origin. Find net force on the rod.
Q.4
A square cardboard of side l and mass m is suspended from a horizontal axis XY as shown in figure. A single wire is wound along the periphery of board and carrying a clockwise current I. At t = 0, a vertical downward magnetic field of induction B is switched on. Find the minimum value of B so that the board will be able to rotate up to horizontal level.
Q.5
A straight segment OC (of length L meter) of a circuit carrying a current I amp is placed along the x-axis. Two infinitely ling straight wires A and B ,each extending form z = – ∞ to + ∞, are fixed at y = – a metre and y = +a metre respectively, as shown in the figure. If the wires A and B each carry a current I amp into plane of the paper. Obtain the expression for the force acting on the segment OC. What will be the force OC if current in the wire B is reversed?
Q.6
A very long straight conductor has a circular cross-section of radius R and carries a current density J. Inside the conductor there is a cylindrical hole of radius a whose axis is parallel to the axis of the conductor and a distance b from it. Let the z-axis be the axis of the conductor, and let the axis of the hole be at x = b. Find the magnetic field on the x = axis at x = 2R on the y = axis at y = 2R.
(a) (b) Q.7
Q charge is uniformly distributed over the same surface of a right circular cone of semi-vertical angle θ and height h. The cone is uniformly rotated about its axis at angular velocity ω. Calculated associated magnetic dipole moment.
9
Q.9
A long straight wire carries a current of 10 A directed along the negative y-axis as shown in figure. A uniform magnetic field B0 of magnitude 10−6 T is directed parallel to the x-axis. What is the resultant magnetic field at the following points? (a) x = 0 , z = 2 m ; (b) x = 2 m, z = 0 ; (c) x = 0 , z = − 0.5 m
Q.10 A stationary, circular wall clock has a face with a radius of 15cm. Six turns of wire are wound around its perimeter, the wire carries a current 2.0 A in the clockwise direction. The clock is located, where there is a constant , uniform external magnetic field of 70 mT (but the clock still keeps perfect time) at exactly 1:00 pm, the hour hand of the clock points in the direction of the external magnetic field (a) After how many minutes will the minute hand point in the direction of the torque on the winding due to the magnetic field ? (b) What is the magnitude of this torque. Q.11
A U-shaped wire of mass m and length l is immersed with its two ends in mercury (see figure). The wire is in a homogeneous field of magnetic induction B. If a charge, that is, a current pulse q = ∫ idt , is sent through the wire, the wire will jump up. Calculate, from the height h that the wire reaches, the size of the charge or current pulse, assuming that the time of the current pulse is very small in comparision with the time of flight. Make use of the fact that impulse of force equals
∫
F dt ,which equals mv. Evaluate q for B = 0.1 Wb/m2, m = 10gm,
= 20cm & h = 3 meters.[g = 10 m/s2] Q.12 A current i, indicated by the crosses in fig. is established in a strip of copper of height h and width w. A uniform field of magnetic induction B is applied at right angles to the strip. (a) Calculate the drift speed vd for the electrons. (b) What are the magnitude and dirction of the magnetic force F acting on the electrons? (c) What would the magnitude & direction of homogeneous electric field E have to be in order to counter balance the effect of the magnetic field ? (d) What is the voltage V necessary between two sides of the conductor in order to create this field E? Between which sides of the conductor would this voltage have to be applied ? (e) If no electric field is applied form the outside the electrons will be pushed somewhat to one side & thereforce will give rise to a uniform electric field EH across the conductor untill the force of this electrostatic field EH balanace the magnetic forces encountered in part (b) . What will be the magnitude and direction of the field EH? Assume that n, the number of conduction electrons per unit volume, is 1.1x1029/m3 & that h = 0.02 meter , w = 0.1cm , i = 50 amp , & B = 2 webers/meter2.
10
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A wire loop carrying current I is placed in the X-Y plane as shown in the figure (a) If a particle with charge +Q and mass m is placed at the centre P and given a velocity along NP (fig). Find its instantaneous acceleration (b) If an external uniform magnetic induction field B = B i is applied, find the torque acting on the loop due to the field.
Page 10 of 20 MAGNETIC EFFECTS OF CURRENT
Q.8
Q.13(a) A rigid circular loop of radius r & mass m lies in the xy plane on a flat table and has a current
I flowing in it. At this particular place , the earth's magnetic field is B = Bx i + By j . How large must I be before one edge of the loop will lift from table ?
(b)
Repeat if, B = Bx i + Bz k .
Q.14 Zeeman effect . In Bohr's theory of the hydrogen atom the electron can be thought of as moving in a circular orbit of radius r about the proton . Suppose that such an atom is placed in a magnetic field, with the plane of the orbit at right angle to B. (a) If the electron is circulating clockwise, as viewed by an observer sighting along B, will the angular frequency increase or decrease? (b) What if the electron is circulating counterclockwise? Assume that the orbit radius does not change. Q.15 In above problem show that the change in frequency of rotation caused by the magnete field is given Be . Such frequency shifts were actually observed by Zeeman in 1896. approximately by ∆v = ± 4πm Q.16 A square loop of wire of edge a carries a current i . (a) Show that B for a point on the axis of the loop and a distance x from its centre is given by,
B= (b) (c)
(
4 µ 0 ia 2
π 4 x2 + a 2
) (4x
2
+ 2a 2
)
1/ 2
.
Can the result of the above problem be reduced to give field at x = 0 ? Does the square loop behave like a dipole for points such that x >> a ? If so , what is its dipole moment?
Q.17 A conductor carrying a current i is placed parallel to a current per unit width j0 and width d, as shown in the figure. Find the force per unit lenght on the coductor. Q.18 Find the work and power required to move the conductor of length l shown in the fig. one full turn in the anticlockwise direction at a rotational frequency of n revolutions per second if the magnetic field is of magnitude B0 everywhere and points radially outwards from Z-axis. The figure shows the surface traced by the wire AB. Q.19 The figure shows a conductor of weight 1.0 N and length L = 0.5 m placed on a rough inclined plane making an angle 300 with the horizontal so that conductor is perpendicular to a uniform horizontal magnetic field of induction B = 0.10 T. The coefficient of static friction between the conductor and the plane is 0.1. A current of I = 10 A flows through the conductor inside the plane of this paper as shown. What is the force needed to be the applied parallel to the inclined plane to sustaining the conductor at rest? Q.20 An electron gun G emits electron of energy 2kev traveling in the (+)ve x-direction. The electron are required to hit the spot S where GS = 0.1m & the line GS makes an angle of 60° with the x-axis, as shown in the fig. A uniform magnetic field B parallel to GS exists in the region outsiees to electron gun. Find the minimum value of B needed to make the electron hit S .
11
EXERCISE # III Q.1
A battery is connected between two points A and B the circumference of a uniform conducting ring of radius r and resistance R . One of the arcs AB of the ring subtends an angle θ at the centre . The value of the magnetic induction at the centre due to the current in the ring is : [ JEE '95, 2] (A) zero , only if θ = 180º (B) zero for all values of θ (C) proportional to 2 (180º - θ) (D) inversely proportional to r
Q.2
Two insulated rings, one slightly smaller diameter than the other, are suspended along their diameter as shown, initially the planes of the rings are mutually perpendicular when a steady current is set up in each of them : [IIT '95, 1] (A) The two rings rotate to come into a common plane (B) The inner ring oscillates about its initially position (C) The outer ring stays stationary while the inner one moves into the plane of the outer ring (D) The inner ring stays stationary while the outer one moves into the plane of the inner ring
Q.3
An electron in the ground state of hydrogen atom is revolving in anticlock-wise direction in a circular orbit of radius R . Obtain an expression for the orbital magnetic dipole moment of the electron The atom is placed in a uniform magnetic. Induction B such that the plane normal of the electron orbit makes an angle of 300 with the magnetic induction. Find the torque experienced by the orbiting electron. [JEE '96, 5]
(i) (ii)
Q.4
A proton , a deuteron and an α-particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field . If rp , rd & rα denote respectively the radii of the trajectories of these [JEE '97, 1] particles then : (A) rα = rp < rd (B) rα > rd > rp (C) rα = rd > rp (D) rp = rd = rα
Q.5
3 infinitely long thin wires each carrying current i in the same direction , are in the x-y plane of a gravity free space . The central wire is along the y-axis while the other two are along x = ± d. Find the locus of the points for which the magnetic field B is zero . If the central wire is displaced along the z-direction by a small amount & released, show that it will execute simple harmonic motion . If the linear density of the wires is λ , find the frequency of oscillation. [JEE '97, 5]
(i) (ii)
Q.6 (i)
Select the correct alternative(s) . [ JEE '98, 2 + 2 + 2 ] Two very long, straight, parallel wires carry steady currents I & − I respectively. The distance between the wires is d. At a certain instant of time, a point charge q is at a point equidistant from the two wires, in the plane of the wires . Its instantaneous velocity v is perpendicular to this plane. The magnitude of the force due to the magnetic field acting on the charge at this instant is : (A)
(ii)
µ 0 Iqv 2 πd
(B)
µ 0 Iqv πd
(C)
2 µ 0 Iqv πd
(D) 0
Let [∈0] denote the dimensional formula of the permittivity of the vaccum and [µ0] that of the permeability of the vacuum . If M = mass, L = length, T = time and I = electric current , (A) [∈0] = M−1 L−3 T2 I (B) [∈0] = M−1 L−3 T4 I2 (C) [µ0] = MLT−2 I−2 (D) [µ0] = ML2 T−1 I
12
(iii)
Two particles, each of mass m & charge q, are attached to the two ends of a light rigid rod of length 2 R . The rod is rotated at constant angular speed about a perpendicular axis passing through its centre. The ratio of the magnitudes of the magnetic moment of the system & its angular momentum about the centre of the rod is : (A)
q 2m
(B)
q m
(C)
2q m
(D)
q πm
Q.7
A particle of mass m & charge q is moving in a region where uniform, constant electric and magnetic fields E & B are present, E & B are parallel to each other. At time t = 0 the velocity v0 of the particle is perpendicular to E . (assume that its speed is always mBvB (C) mA < mB and vA < vB (D) mA = mB and vA = vB [JEE, 2001 (Scr)] Q.15 A non-planar loop of conducting wire carrying a current I is placed as shown in the figure. Each of the straight sections of the loop is of length 2a. The magnetic field due to this loop at the point P (a, 0, a) points in the direction 1 ˆ ˆ ˆ 1 (− j + k + i ) (−ˆj + kˆ ) (B) (A) 3 2 1 ˆ ˆ ˆ 1 ˆ ˆ (i + j + k ) (i + k ) (C) (D) [JEE, 2001 (Scr)] 2 3
14
Q.16 A coil having N turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. When a current I passes through the coil, the magnetic field at the centre is [JEE, 2001 Screening] (A)
µ 0 NI b
(B)
2µ 0 NI a
(C)
µ 0 NI
2( b − a )
ln
b a
(D)
µ0I N 2( b − a )
ln
b a
Q.17 A particle of mass m and charge q moves with a constant velocity v along the positive x direction. It enters a region containing a uniform magnetic field B directed along the negative z direction, extending from x = a to x = b. The minimum value of v required so that the particle can just enter the region x > b is (A) q b B/m (B) q( b – a) B/m (C) q a B/m (D) q(b + a) B/2m [JEE 2002 (screening), 3] Q.18 A long straight wire along the z-axis carries a current I in the negative z direction. The magnetic vector [JEE 2002 (screening), 3] field B at a point having coordinates (x, y) in the z = 0 plane is µ 0 I (xi + yj ) µ 0 I (xj − yi ) µ 0 I (xi − yj ) µ 0 I (yi − xj) (A) (C) (B) (D) 2 2 2 2 2 2 2 π (x + y ) 2 π (x 2 + y 2 ) 2π (x + y ) 2π (x + y ) Q.19 The magnetic field lines due to a bar magnet are correctly shown in
(A)
(B)
(C)
[JEE 2002 (screening), 3]
(D)
Q.20 A rectangular loop PQRS made from a uniform wire has length a, width b and mass m. It is free to rotate about the arm PQ, which remains hinged along a horizontal line taken as the y-axis (see figure). Take the vertically upward direction as the z-axis. A uniform magnetic field B = (3 i + 4 k ) B 0
(a) (b) (c)
exists in the region. The loop is held in the x-y plane and a current I is passed through it. The loop is now released and is found to stay in the horizontal position in equilibrium. What is the direction of the current I in PQ? Find the magnetic force on the arm RS. Find the expression for I in terms of B0, a, b and m. [JEE 2002, 1+1+3]
Q.21 A circular coil carrying current I is placed in a region of uniform magnetic field acting perpendicular to a coil as shown in the figure. Mark correct option [JEE 2003 (Scr)] (A) coil expands (B) coil contracts (C) coil moves left (D) coil moves right
Q.22 Figure represents four positions of a current carrying coil is a magnetic field directed towards right. nˆ represent the direction of area of vector of the coil. The correct order of potential energy is : [JEE 2003 (Scr)] (A) I > III > II > IV (B) I < III < II < IV (C) IV < I < II < II (D) II > II > IV > I
15
Q.23 A wheel of radius R having charge Q, uniformly distributed on the rim of the wheel is free to rotate about a light horizontal rod. The rod is suspended by light inextensible stringe and a magnetic field B is applied as shown in the figure. The 3T initial tensions in the strings are T0. If the breaking tension of the strings are 0 , 2 find the maximum angular velocity ω0 with which the wheel can be rotate. [JEE 2003] Q.24 A proton and an alpha particle, after being accelerated through same potential difference, enter a uniform magnetic field the direction of which is perpendicular to their velocities. Find the ratio of radii of the circular paths of the two particles. [JEE 2004] Q.25
(a) (b) (c)
In a moving coil galvanometer, torque on the coil can be expressed as τ = ki, where i is current through the wire and k is constant. The rectangular coil of the galvanometer having numbers of turns N, area A and moment of inertia I is placed in magnetic field B. Find k in terms of given parameters N, I, A and B. the torsional constant of the spring, if a current i0 produces a deflection of π/2 in the coil in reaching equilibrium position. the maximum angle through which coil is deflected, id charge Q is passed through the coil almost instantaneously. (Ignore the damping in mechanical oscillations) [JEE 2005]
Q.26 An infinite current carrying wire passes through point O and in perpendicular to the plane containing a current carrying loop ABCD as shown in the figure. Choose the correct option (s). (A) Net force on the loop is zero. (B) Net torque on the loop is zero. (C) As seen from O, the loop rotates clockwise. (D) As seen from O, the loop rotates anticlockwise
16
[JEE 2006]
ANSWER KEY EXERCISE # I Q.1
2 µ 0i 8π l
Q.4
zero
Q.7
B=
Q.10
µ 0iqv 2πa
µ0 i 4πR
Q.3
Q.5
µ0I 3 ˆ 1 ˆ k + j 4R 4 π
Q.6
(
)
µ 0 br12
Q.21
t= m
Q.22
mEI Be
3
, B2 =
(2 2 − 1) µ 0i πa µ0 i 3 π + 1 4πr 2
Q.8
(i) 1.3 ×10–4T, (ii) zero Q.9
Q.12
2mv 0 qB
Q.11
10 kˆ
Q.15
i1 = 0.1110 A, i2 = 0.096 A
3mv 2 3mv 3 , (b) , (c) zero (a) 4qa 4a
Q.19 B1 =
Q.24
5 4π × 10–5 T 2 2
2 2 π2 − 2 π + 1
Q.14 F = αa2i ˆj
Q.17
Q.2
Q.13 zero
Q.16
4
µ 0 bR 3
Q.20 T0 = 2 π
3r2
m = 0.57 s 6IB
Q.23
µ 0 I I′ C 1 1 − to the left 2π a b
Q.25
W µ 0 I1I 2 r2 = n = 27.6 µ J/m 2π r1
EXERCISE # II Q.1
µ0I0 3b 2 2 π (a + b 2 )
Q.2
7
Q.18 In the plane of the drawing from right to left
dB q α , where α = sin–1 2 mV qB
2 IRB
µ0 weber.m–1
(i)
µ 0 4I along Y-axis, 4π a
µ0 I2 1 (ii) 4π 2a 10 , tan 4 + π with positive axis 3
17
Q.3
µ 0 I1I 2 ln (3) along – ve z direction 4π
Q.5
µ0 I2 L2 + a 2 n −kˆ , zero F= 2 π a2
Q.6
1 µ 0 J a 2 b a2 µ0 J a 2 R − µ J R − − (a) B = 2 2 R − b 2 , (b) Bx = 0 4 , By = 2 4R 2 + b 2 4R 2 + b 2
Q.7
Qω 2 h tan 2 θ 4
Q.9
(a) 0 (b) 1.41 × 10 –6 T , 45º in xz - plane, (c) 5 × 10 –6 T , + x - direction]
mg 2 Il
Q.4
( )
Q.8
QV µ 0 I 3 3 π 3 2 (a) m 6a π −1 , (b) τ=BI − a ˆj 3 4
Q.10 (a) 20 min. (b) 5.94 x 10–2 Nm Q.11
15 C
Q.12 (a) 1.4 x 10−4 m/s (b) 4.5 x 10−23 N (down) (c) 2.8 x 10−4 V/m (down) (d) 5.7 x 10−6 V (top + , bottom −) (e) same as (c) Q.13 (a) I =
Q.17
mg
πr
(
B2x
+
B2y
)
1/ 2
(b) I =
mg π r Bx
µ 0 iJ 0
d tan −1 (−kˆ ) π 2h
Q.14 (a) increase, (b) decrease
Q.18 − 2 π r B0 i l , − 2 π r B0 i l n Q.20 Bmin = 4.7×10–3 T
Q.19 0.62 N < F < 0.88 N
EXERCISE # III Q.1 B
Q.5
ehB eh Q.3 (i) m = 4πm ; τ = 8πm
Q.2 A
z=0,x=±
d I , (ii) 2πd 3
µ0 πλ
Q.4 A
Q.6 (i) D (ii) B, C (iii) A
q qB t + v 0 cos ωt + [v0 sin ωt] k , where ω = ; kˆ = ( v 0 x E )/ v 0 x E Q.7 v = E m m
Q.8 (a) τ =
BI 0 L2 ˆi −ˆj 2
( )
(b) θ =
3 BI 0 ∆t 2 4 M
Q.9 A
18
mv0 πm Q.10 (a) 2qB (b)velocity=-v, time= qB 0 0
Q.12 (i)
Q.11 (i) C
(ii) B (iii) C
(iv) C
µ0 I − 4R q v0 kˆ (ii) F1 = 2 I R B F2 =2 I R B , Net force = F1 + F2 = 4 I R B i
Q.13 (a) 6.6 × 10–5T, (b) 0, 0, 8 × 10–6Nt Q.14 B
Q.15 D
Q.16 C
Q.17 B
Q.18 A
Q.19 D
mg Q.20 (a) current in loop PQRS is clockwise from P to QRS., (b) F = BI0b (3kˆ−4ˆi) , (c) I = 6bB0
Q.21 A
Q.25
(a) k = NAB, (b) C =
d T0
Q.23 ω =
Q.22 A
2i 0 NAB π
, (c) Q ×
QR 2 B
NABπ 2li 0
19
rp mp qα 1 Q.24 r = m . q = 2 α α p
Q.26 A,C
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: PHYSICS TOPIC: XII P6. Electromagnetic Induction and Alternating Current Index: 1. Key Concepts 2. Exercise I 3. Exercise II 4. Exercise III 5. Exercise IV 6. Answer Key 7. 34 Yrs. Que. from IIT-JEE 8. 10 Yrs. Que. from AIEEE
1
When a conductor is moved across a magnetic field, an electromotive force (emf) is produced in the conductor. If the conductors forms part of a closed circuit then the emf produced caused an electric current to flow round the circuit. Hence an emf (and thus a current) is induced in the conductor as a result of its movement across the magnetic field. This is known as "ELECTROMAGNETIC INDUCTION." 1.
MAGNETIC FLUX : φ = B . A = BA cos θ weber for uniform B . φ = ∫ B . d A for non uniform B .
2. (i)
FARADAY'S LAWS OF ELECTROMAGNETIC INDUCTION : An induced emf is setup whenever the magnetic flux linking that circuit changes.
(ii)
The magnitude of the induced emf in any circuit is proportional to the rate of change of the magnetic dφ flux linking the circuit, ε α . dt
3.
LENZ'S LAWS : The direction of an induced emf is always such as to oppose the cause producing it .
4.
LAW OF EMI : dφ . The negative sign indicated that the induced emf opposes the change of the flux . e=− dt
5.
EMF INDUCED IN A STRAIGHT CONDUCTOR IN UNIFORM MAGNETIC FIELD : E = BLV sin θ volt where B = flux density in wb/m2 ; L = length of the conductor (m) ; V = velocity of the conductor (m/s) ; θ = angle between direction of motion of conductor & B .
6.
COIL ROTATION IN MAGNETIC FIELD SUCH THAT AXIS OF ROTATION IS PERPENDICULAR TO THE MAGNETIC FIELD : Instantaneous induced emf . E = NABω sin ωt = E0 sin ωt , where N = number of turns in the coil ; A = area of one turn ; B = magnetic induction ; ω = uniform angular velocity of the coil ; E0 = maximum induced emf .
7.
SELF INDUCTION & SELF INDUCTANCE : When a current flowing through a coil is changed the flux linking with its own winding changes & due to the change in linking flux with the coil an emf is induced which is known as self induced emf & this phenomenon is known as self induction . This induced emf opposes the causes of Induction. The property of the coil or the circuit due to which it opposes any change of the current coil or the circuit is known as SELF & INDUCTANCE . It's unit is Henry . φ φs = Li Coefficient of Self inductance L = s or i
2
Page 2 of 16 E.M.I. & A.C.
KEY CONCEPTS
(i)
shape of the loop
&
(ii)
medium i = current in the circuit . φs = magnetic flux linked with the circuit due to the current i . dφ s di d self induced emf es = =− (Li) = − L (if L is constant) dt dt dt
8.
MUTUAL INDUCTION : If two electric circuits are such that the magnetic field due to a current in one is partly or wholly linked with the other, the two coils are said to be electromagnetically coupled circuits . Than any change of current in one produces a change of magnetic flux in the other & the latter opposes the change by inducing an emf within itself . This phenomenon is called MUTUAL INDUCTION & the induced emf in the latter circuit due to a change of current in the former is called MUTUALLY INDUCED EMF . The circuit in which the current is changed, is called the primary & the other circuit in which the emf is induced is called the secondary. The co0efficient of mutual induction (mutual inductance) between two electromagnetically coupled circuit is the magnetic flux linked with the secondary per unit current in the primary. φ m flux linked with sec ondary Mutual inductance = M = = mutually induced emf . Ip current in the primary d dI dφm =− (MI) = − M (If M is constant) dt dt dt M depends on (1) geometery of loops (2) medium (3) orientation & distance of loops .
Em =
9.
SOLENOID : There is a uniform magnetic field along the axis the solenoid (ideal : length >> diameter) B = µ ni where ; µ = magnetic permeability of the core material ; n = number of turns in the solenoid per unit length ; i = current in the solenoid ; Self inductance of a solenoid L = µ0 n2Al ; A = area of cross section of solenoid .
10.
SUPER CONDUCTION LOOP IN MAGNETIC FIELD : R = 0 ; ε = 0. Therefore φtotal = constant. Thus in a superconducting loop flux never changes. (or it opposes 100%)
11.
(i)
(ii)
ENERGY STORED IN AN INDUCTOR : 1 2 W= LI . 2 Energy of interation of two loops U = I1φ2 = I2φ1 = MI1I2 , where M is mutual inductance .
3
Page 3 of 16 E.M.I. & A.C.
L depends only on ;
GROWTH OF A CURRENT IN AN L − R CIRCUIT : E I= (1 − e−Rt/L) . [ If initial current = 0 ] R L = time constant of the circuit . R E I0 = . R (i) L behaves as open circuit at t = 0 [ If i = 0 ] L behaves as short circuit at t = ∞ always . L Large Curve (1) → R L Curve (2) → Small R DECAY OF CURRENT : Initial current through the inductor = I0 ; (ii)
13.
Current at any instant i = I0e−Rt/L
4
Page 4 of 16 E.M.I. & A.C.
12.
Q.1
The horizontal component of the earth’s magnetic field at a place is 3 × 10–4 T and the dip is tan–1(4/3). A metal rod of length 0.25 m placed in the north-south position is moved at a constant speed of 10cm/s towards the east. Find the e.m.f. induced in the rod.
Q.2
A wire forming one cycle of sine curve is moved in x-y plane with velocity V = Vx i + Vy j . There exist a magnetic field B = − B 0 k . Find the motional emf develop across the ends PQ of wire.
Q.3
A conducting circular loop is placed in a uniform magnetic field of 0.02 T, with its plane perpendicular to the field . If the radius of the loop starts shrinking at a constant rate of 1.0 mm/s, then find the emf induced in the loop, at the instant when the radius is 4 cm.
Q.4
Find the dimension of the quantity
Q.5
A rectangular loop with a sliding connector of length l = 1.0 m is situated in a uniform magnetic field B = 2T perpendicular to the plane of loop. Resistance of connector is r = 2Ω. Two resistances of 6Ω and 3Ω are connected as shown in figure. Find the external force required to keep the connector moving with a constant velocity v = 2m/s.
Q.6
Two concentric and coplanar circular coils have radii a and b(>>a)as shown in figure. Resistance of the inner coil is R. Current in the outer coil is increased from 0 to i , then find the total charge circulating the inner coil.
Q.7
A horizontal wire is free to slide on the vertical rails of a conducting frame as shown in figure. The wire has a mass m and length l and the resistance of the circuit is R. If a uniform magnetic field B is directed perpendicular to the frame, then find the terminal speed of the wire as it falls under the force of gravity.
Q.8
A metal rod of resistance 20Ω is fixed along a diameter of a conducting ring of radius 0.1 m and lies on x-y plane. There is a magnetic field B = (50T) kˆ . The ring rotates with an angular velocity ω = 20 rad/sec about its axis. An external resistance of 10Ω is connected across the centre of the ring and rim. Find the current through external resistance.
Q.9
In the given current, find the ratio of i1 to i2 where i1 is the initial (at t = 0) current and i2 is steady state (at t = ∞) current through the battery.
L , where symbols have usual meaining. RCV
Q.10 In the circuit shown, initially the switch is in position 1 for a long time. Then the switch is shifted to position 2 for a long time. Find the total heat produced in R2.
5
Page 5 of 16 E.M.I. & A.C.
EXERCISE–I
Two resistors of 10Ω and 20Ω and an ideal inductor of 10H are connected to a 2V battery as shown. The key K is shorted at time t = 0. Find the initial (t = 0) and final (t → ∞) currents through battery.
Q.12
There exists a uniform cylindrically symmetric magnetic field directed along the axis of a cylinder but varying with time as B = kt. If an electron is released from rest in this field at a distance of ‘r’ from the axis of cylinder, its acceleration, just after it is released would be (e and m are the electronic charge and mass respectively)
Q.13 An emf of 15 volt is applied in a circuit containing 5 H inductance and 10 Ω resistance. Find the ratio of the currents at time t = ∞ and t = 1 second. Q.14
A uniform magnetic field of 0.08 T is directed into the plane of the page and perpendicular to it as shown in the figure. A wire loop in the plane of the page has constant area 0.010 m2. The magnitude of magnetic field decrease at a constant rate of 3.0 × 10–4 Ts–1. Find the magnitude and direction of the induced emf in the loop.
Q.15 In the circuit shown in figure switch S is closed at time t = 0. Find the charge which passes through the battery in one time constant.
Q.16 Two coils, 1 & 2, have a mutual inductance = M and resistances R each. A current flows in coil 1, which varies with time as: I1 = kt2 , where K is a constant and 't' is time. Find the total charge that has flown through coil 2, between t = 0 and t = T. Q.17 In a L–R decay circuit, the initial current at t = 0 is I. Find the total charge that has flown through the resistor till the energy in the inductor has reduced to one–fourth its initial value. Q.18 A charged ring of mass m = 50 gm, charge 2 coulomb and radius R = 2m is placed on a smooth horizontal surface. A magnetic field varying with time at a rate of (0.2 t) Tesla/sec is applied on to the ring in a direction normal to the surface of ring. Find the angular speed attained in a time t1 = 10 sec. Q.19 A capacitor C with a charge Q0 is connected across an inductor through a switch S. If at t = 0, the switch is closed, then find the instantaneous charge q on the upper plate of capacitor. Q.20 A uniform but time varying magnetic field B = Kt – C ; (0 ≤ t ≤ C/K), where K and C are constants and t is time, is applied perpendicular to the plane of the circular loop of radius ‘a’ and resistance R. Find the total charge that will pass around the loop. Q.21 A coil of resistance 300Ω and inductance 1.0 henry is connected across an alternating voltage of frequency 300/2π Hz. Calculate the phase difference between the voltage and current in the circuit. Q.22 Find the value of an inductance which should be connected in series with a capacitor of 5 µF, a resistance of 10Ω and an ac source of 50 Hz so that the power factor of the circuit is unity.
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Page 6 of 16 E.M.I. & A.C.
Q.11
Q.24 When 100 volt D.C. is applied across a coil, a current of one ampere flows through it, when 100 V ac of 50 Hz is applied to the same coil, only 0.5 amp flows. Calculate the resistance and inductance of the coil. Q.25 A 50W, 100V lamp is to be connected to an ac mains of 200V, 50Hz. What capacitance is essential to be put in seirs with the lamp. List of recommended questions from I.E. Irodov. 3.288 to 3.299, 3.301 to 3.309, 3.311, 3.313, 3.315, 3.316, 3.326 to 3.329, 3.331, 3.333 to 3.335, 4.98, 4.99, 4.100, 4.134, 4.135, 4.121, 4.124, 4.125, 4.126, 4.136, 4.137, 4.141, 4.144
7
Page 7 of 16 E.M.I. & A.C.
Q.23 In an L-R series A.C circuit the potential difference across an inductance and resistance joined in series are respectively 12 V and 16V. Find the total potential difference across the circuit.
Q.1
(i) (ii) (iii)
Two straight conducting rails form a right angle where their ends are joined. A conducting bar contact with the rails starts at vertex at the time t = 0 & moves symmetrically with a constant velocity of 5.2 m/s to the right as shown in figure. A 0.35 T magnetic field points out of the page. Calculate: The flux through the triangle by the rails & bar at t = 3.0 s. The emf around the triangle at that time. In what manner does the emf around the triangle vary with time .
Q.2
Two long parallel rails, a distance l apart and each having a resistance λ per unit length are joined at one end by a resistance R. A perfectly conducting rod MN of mass m is free to slide along the rails without friction. There is a uniform magnetic field of induction B normal to the plane of the paper and directed into the paper. A variable force F is applied to the rod MN such that, as the rod moves, a constant current i flows through R. Find the velocity of the rod and the applied force F as function of the distance x of the rod from R
Q.3
A wire is bent into 3 circular segments of radius r = 10 cm as shown in figure . Each segment is a quadrant of a circle, ab lying in the xy plane, bc lying in the yz plane & ca lying in the zx plane. if a magnetic field B points in the positive x direction, what is the magnitude of the emf developed in the wire, when B increases at the rate of 3 mT/s ? what is the direction of the current in the segment bc.
(i)
(ii) Q.4
(i) (ii) (iii) Q.5
(i) (ii) Q.6
Consider the possibility of a new design for an electric train. The engine is driven by the force due to the vertical component of the earths magnetic field on a conducting axle. Current is passed down one coil, into a conducting wheel through the axle, through another conducting wheel & then back to the source via the other rail. what current is needed to provide a modest 10 − KN force ? Take the vertical component of the earth's field be 10 µT & the length of axle to be 3.0 m . how much power would be lost for each Ω of resistivity in the rails ? is such a train unrealistic ? A square wire loop with 2 m sides in perpendicular to a uniform magnetic field, with half the area of the loop in the field . The loop contains a 20 V battery with negligible internal resistance. If the magnitude of the field varies with time according to B = 0.042 − 0.87 t, with B in tesla & t in sec. What is the total emf in the circuit ? What is the direction of the current through the battery ? A rectangular loop of dimensions l & w and resistance R moves with constant velocity V to the right as shown in the figure. It continues to move with same speed through a region containing a uniform magnetic field B directed into the plane of the paper & extending a distance 3 W. Sketch the flux, induced emf & external force acting on the loop as a function of the distance.
8
Page 8 of 16 E.M.I. & A.C.
EXERCISE–II
A rectangular loop with current I has dimension as shown in figure . Find the magnetic flux φ through the infinite region to the right of line PQ.
Q.8
A square loop of side 'a' & resistance R moves with a uniform velocity v away from a long wire that carries current I as shown in the figure. The loop is moved away from the wire with side AB always parallel to the wire. Initially, distance between the side AB of the loop & wire is 'a'. Find the work done when the loop is moved through distance 'a' from the initial position.
Q.9
Two long parallel conducting horizontal rails are connected by a conducting wire at one end. A uniform magnetic field B exists in the region of space. A light uniform ring of diameter d which is practically equal to separation between the rails, is placed over the rails as shown in the figure. If resistance of ring is λ per unit length, calculate the force required to pull the ring with uniform velocity v.
Q.10 A long straight wire is arranged along the symmetry axis of a toroidal coil of rectangular cross−section, whose dimensions are given in the figure. The number of turns on the coil is N, and permeability of the surrounding medium is unity. Find the amplitude of the emf induced in this coil, if the current i = im cos ωt flows along the straight wire. Q.11
A uniform magnetic field B fills a cylindrical volumes of radius R. A metal rod CD of length l is placed inside the cylinder along a chord of the circular cross-section as shown in the figure. If the magnitude of magnetic field increases in the direction of field at a constant rate dB/dt, find the magnitude and direction of the EMF induced in the rod.
Q.12 A variable magnetic field creates a constant emf E in a conductor ABCDA. The resistances of portion ABC, CDA and AMC are R1, R2 and R3 respectively. What current will be shown by meter M? The magnetic field is concentrated near the axis of the circular conductor.
Q.13 In the circuit shown in the figure the switched S1 and S2 are closed at time t = 0. After time t = (0.1) ln 2 sec, switch S2 is opened. Find the current in the circuit at time t = (0.2) ln 2 sec.
Q.14 (i) (ii) (iii) (iv)
Find the values of i1 and i2 immediately after the switch S is closed. long time later, with S closed. immediately after S is open. long time after S is opened.
9
Page 9 of 16 E.M.I. & A.C.
Q.7
Q.16 Suppose the emf of the battery, the circuit shown varies with time t so the current is given by i(t) = 3 + 5t, where i is in amperes & t is in seconds. Take R = 4Ω, L = 6H & find an expression for the battery emf as function of time. Q.17 A current of 4 A flows in a coil when connected to a 12 V dc source. If the same coil is connected to a 12V, 50 rad/s ac source a current of 2.4 A flows in the circuit. Determine the inductance of the coil. Also find the power developed in the circuit if a 2500 µF capacitor is connected in series with the coil. Q.18 An LCR series circuit with 100Ω resistance is connected to an ac source of 200 V and angular frequency 300 rad/s. When only the capacitance is removed, the current lags behind the voltage by 60°. When only the inductance is removed, the current leads the voltage by 60°. Calculate the current and the power dissipated in the LCR circuit. Q.19 A box P and a coil Q are connected in series with an ac source of variable frequency. The emf of source at 10 V. Box P contains a capacitance of 1µF in series with a resistance of 32Ω coil Q has a self-inductance 4.9 mH and a resistance of 68Ω series. The frequency is adjusted so that the maximum current flows in P and Q. Find the impedance of P and Q at this frequency. Also find the voltage across P and Q respectively. Q.20 A series LCR circuit containing a resistance of 120Ω has angular resonance frequency 4 × 105 rad s–1. At resonance the voltages across resistance and inductance are 60 V and 40 V respectively. Find the values of L and C. At what frequency the current in the circuit lags the voltage by 45°?
10
Page 10 of 16 E.M.I. & A.C.
Q.15 Consider the circuit shown in figure. The oscillating source of emf deliver a sinusoidal emf of amplitude emax and frequency ω to the inductor L and two capacitors C1 and C2. Find the maximum instantaneous current in each capacitor.
EXERCISE–III
Q.2
A rectangular frame ABCD made of a uniform metal wire has a straight connection between E & F made of the same wire as shown in the figure. AEFD is a square of side 1 m & EB = FC = 0.5 m. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the place of the paper & normal to it . The rate of change of the magnetic field is 1 T/s, the resistance per unit length of the wire is 1 Ω/m. Find the current in segments AE, BE & EF. [JEE '93, 5] An inductance L, resistance R, battery B and switch S are connected in series. Voltmeters VL and VR are connected across L and R respectively. When switch is closed: (A) The initial reading in VL will be greater than in VR. (B) The initial reading in VL will be lesser than VR. (C) The initial readings in VL and VR will be the same. (D) The reading in VL will be decreasing as time increases.
Page 11 of 16 E.M.I. & A.C.
Q.1
[JEE’93, 2] Q.3
Two parallel vertical metallic rails AB & CD are separated by 1 m. They are connected at the two ends by resistance R1 & R2 as shown in the figure. A horizontally metallic bar L of mass 0.2 kg slides without friction, vertically down the rails under the action of gravity. There is a uniform horizontal magnetic field of 0.6T perpendicular to the plane of the rails, it is observed that when the terminal velocity is attained, the power dissipated in R1 & R2 are 0.76 W & 1.2 W respectively. Find the terminal velocity of bar L & value R1 & R2. [ JEE '94, 6]
Q.4
Two different coils have self inductance 8mH and 2mH. The current in one coil is increased at a constant rate. The current in the second coild is also increased at the same constant. At a certain instant of time, the power given to the two coils is the same. At that time the current, the induced voltage and the energy stored in the first coil are I1, V1 and W1 respectively. Corresponding values for the second coil at the [JEE’94, 2] same instant are I2, v2 and W2 respectively. Then: (A)
Q.5
I1 1 = I2 4
(B)
I1 =4 I2
W2 (C) W = 4 1
A metal rod OA of mass m & length r is kept rotating with a constant angular speed ω in a vertical plane about a horizontal axis at the end O. The free end A is arranged to slide without friction along a fixed conducting circular ring in the same plane as that of rotation. A uniform & constant
(a) (b)
V2 1 (D) V = 4 1
magnetic induction B is applied perpendicular & into the plane of rotation as shown in figure. An inductor L and an external resistance R are connected through a switch S between the point O & a point C on the ring to form an electrical circuit. Neglect the resistance of the ring and the rod. Initially, the switch is open. What is the induced emf across the terminals of the switch ? (i) Obtain an expression for the current as a function of time after switch S is closed. (ii) Obtain the time dependence of the torque required to maintain the constant angular speed, given that the rod OA was along the positive X-axis at t = 0. [JEE '95, 10]
11
A solenoid has an inductance of 10 Henry & a resistance of 2 Ω. It is connected to a 10 volt battery. How long will it take for the magnetic energy to reach 1/4 of its maximum value ? [JEE '96, 3]
Q.7
Select the correct alternative. A thin semicircular conducting ring of radius R is falling with its plane vertical in a horizontal magnetic induction B . At the position MNQ the speed of the ring is v & the potential difference developed across the ring is : (A) zero (C) π RBV & Q is at higher potential
Bv π R 2 & M is at higher potential 2 (D) 2 RBV & Q is at higher potential
(B)
[JEE'96, 2] Q.8
Fill in the blank. A metallic block carrying current I is subjected to a uniform magnetic induction B j . The moving charges experience a force F given by ______ which results in the lowering of the potential of the face ______. [assume the speed of the carrier to be v]
Q.9
(i) (ii)
[JEE '96, 2]
A pair of parallel horizontal conducting rails of negligible resistance shorted at one end is fixed on a table. The distance between the rails is L. A conducting massless rod of resistance R can slide on the rails frictionlessly. The rod is tied to a massless string which passes over a pulley fixed to the edge of the table. A mass m, tied to the other end of the string hangs vertically. A constant magnetic field B exists perpendicular to the table. If the system is released from rest, calculate: the terminal velocity achieved by the rod. the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity. [JEE '97, 5]
Q.10 A current i = 3.36 (1 + 2t) × 10−2 A increases at a steady rate in a long straight wire. A small circular loop of radius 10−3 m is in the plane of the wire & is placed at a distance of 1 m from the wire. The resistance of the loop is 8.4 x 10−2 Ω. Find the magnitude & the direction of the induced current in the loop. [REE '98, 5] [ JEE '98, 3 × 2 = 6 ,4×2=8] Q.11 Select the correct alternative(s). (i) The SI unit of inductance, the Henry, can be written as : (A) weber/ampere (B) volt − second/ampere (C) joule/(ampere)2 (D) ohm − second (ii)
A small square loop of wire of side l is placed inside a large square loop of wire of side L(L >> l). The loop are co-planar & their centres coincide. The mutual inductance of the system is proportional to :
L 2 L2 (B) (C) (D) L L A metal rod moves at a constant velocity in a direction perpendicular to its length . A constant, uniform magnetic field exists in space in a direction perpendicular to the rod as well as its velocity. Select the correct statement(s) from the following (A) the entire rod is at the same electric potential (B) there is an electric field in the rod (C) the electric potential is highest at the centre of the rod & decreases towards its ends (D) the electric potential is lowest at the centre of the rod & increases towards its ends. (A)
(iii)
12
Page 12 of 16 E.M.I. & A.C.
Q.6
An inductor of inductance 2.0mH,is connected across a charged capacitor of capacitance 5.0µF,and the resulting LC circuit is set oscillating at its natural frequency. Let Q denote the instantaneous charge on the capacitor, and I the current in the circuit .It is found that the maximum value of Q is 200 µC.
(a) (b) (c)
when Q=100µC,what is the value of dI / dt ? when Q=200 µC ,what is the value of I ? Find the maximum value of I.
(d)
when I is equal to one half its maximum value, what is the value of Q
Q.12 Two identical circular loops of metal wire are lying on a table without touching each other. Loop-A carries a current which increases with time. In response, the loop-B [JEE ’99] (A) remains stationary (B) is attracted by the loop-A (C) is repelled by the loop-A (D) rotates about its CM, with CM fixed Q.13 A coil of inductance 8.4 mH and resistance 6Ω is connected to a 12V battery. The current in the coil is 1.0 A at approximately the time (A) 500 s (B) 20 s (C) 35 ms (D) 1 ms [ JEE ’99 ] Q.14 A circular loop of radius R, carrying current I, lies in x-y plane with its centre at origin. The total magnetic flux through x-y plane is (A) directly proportional to I (B) directly proportional to R (C) directly proportional to R2 (D) zero [JEE ’99] Q.15 A magnetic field B = (B0y / a) k is into the plane of paper in the +z direction. B0 and a are positive constants. A square loop EFGH of side a, mass m and resistance R, in x-y plane, starts falling under the influence of gravity. Note the directions of x and y axes in the figure. Find (a) the induced current in the loop and indicate its direction, (b) the total Lorentz force acting on the loop and indicate its direction, (c) an expression for the speed of the loop, v(t) and its terminal value.
[JEE ’99]
Q.16 Two circular coils can be arranged in any of the three situations shown in the figure. Their mutual inductance will be (A) maximum in situation (a) (B) maximum in situation (b) (C) maximum in situation (c) (D) the same in all situations [JEE ’2001, (Scr)] Q.17 An inductor of inductance L = 400 mH and resistors of resistances R1 = 2Ω and R2 = 2Ω are connected to a battery of e.m.f. E = 12V as shown in the figure. The internal resistance of the battery is negligible. The switch S is closed at time t = 0. What is the potential drop across L as a function of time? After the steady state is reached, the switch is opened. What is the direction and the magnitude of current through R1 as a function of time? [JEE ’2001]
13
Page 13 of 16 E.M.I. & A.C.
(iv)
Q.19 A short -circuited coil is placed in a time varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled and the wire radius halved, the electrical power dissipated would be [JEE 2002(Scr), 3] (A) halved (B) the same (C) doubled (D) quadrupled Q.20 A square loop of side 'a' with a capacitor of capacitance C is located between two current carrying long parallel wires as shown. The value of I in the is given as I = I0sinωt. calculate maximum current in the square loop. (a) (b) Draw a graph between charge on the lower plate of the capacitor v/s time.
[JEE 2003]
Q.21 The variation of induced emf (ε) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity is best represented as (A)
(B)
(C)
(D) [JEE 2004(Scr)]
Q.22 In an LR series circuit, a sinusoidal voltage V = Vo sin ωt is applied. It is given that L = 35 mH, R = 11 Ω,
V
ω = 50 Hz and π = 22/7. Find 2π the amplitude of current in the steady state and obtain the phase difference between the current and the voltage. Also plot the variation of current for one cycle on the given graph. [JEE 2004]
O
Vrms = 220
V,
T/4
T/2
3T/4
T
t
Q.23 An infinitely long cylindrical conducting rod is kept along + Z direction. A constant magnetic field is also present in + Z direction. Then current induced will be (A) 0 (B) along +z direction (C) along clockwise as seen from + Z (D) along anticlockwise as seen from + Z [JEE’ 2005 (Scr)] Q. 24 A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R, thickness d (d 0, there is no exchange of energy between the inductor and capacitor (D) at any time t > 0, instantaneous current in the circuit may V
C L
[JEE 2006]
Q.28 If the total charge stored in the LC circuit is Q0, then for t ≥ 0 t π (A) the charge on the capacitor is Q = Q 0 cos + LC 2 t π (B) the charge on the capacitor is Q = Q 0 cos − LC 2 (C) the charge on the capacitor is Q = − LC (D) the charge on the capacitor is Q = −
d 2Q dt 2
1 d 2Q LC dt 2
[JEE 2006]
15
Page 15 of 16 E.M.I. & A.C.
Q.25 In the given diagram, a line of force of a particular force field is shown. Out of the following options, it can never represent (A) an electrostatic field (B) a magnetostatic field (C) a gravitational field of a mass at rest (D) an induced electric field [JEE 2006]
Q.29 What is the advantage of the train? (A) Electrostatic force draws the train (C) Electromagnetic force draws the train
[JEE 2006] (B) Gravitational force is zero. (D) Dissipative force due to friction are absent
Q.30 What is the disadvantage of the train? (A) Train experience upward force due to Lenz's law. (B) Friction force create a drag on the train. (C) Retardation (D) By Lenz's law train experience a drag
[JEE 2006]
Q.31 Which force causes the train to elevate up (A) Electrostatic force (C) magnetic force
[JEE 2006]
Q.32 Match the following Columns Column 1 (A) Dielectric ring uniformly charged (B) Dielectric ring uniformly charged rotating with angular velocity . (C) Constant current in ring i0 0 cos ωt in ring ( D
)
C
u r r e n t
i
=
i
(B) Time varying electric field (D) Induced electric field
Column 2 (P) Time independent electrostatic field out of system (Q) Magnetic field
(R) Induced electric field (S) Magnetic moment
16
[JEE 2006]
Page 16 of 16 E.M.I. & A.C.
Comprehension –IV Magler Train: This train is based on the Lenz law and phenomena of electromagnetic induction. In this there is a coil on a railway track and magnet on the base of train. So as train is deviated then as is move down coil on track repel it and as it move up then coil attract it. Disadvantage of magler train is that as it slow down the forces decreases and as it moves forward so due to Lenz law coil attract it backward. Due to motion of train current induces in the coil of track which levitate it.
ANSWER KEY Q.1
10 µV
Q.6
µ 0ia 2 π 2Rb
Q.2
λVyB0
Q.7
mgR B2l 2
Q.3
5.0 µV
Q.8
1 A 3
Q.4
Q.9
I–1
Q.5
0.8
LE 2 Q.10 2R 12
Q.11
1 1 A, A 15 10
Q.12
erk directed along tangent to the circle of radius r, whose centre lies on the axis of cylinder.. 2m
e2 Q.13 2 e −1
Q.14 3µV, clockwise
Q.15
EL eR 2
1 π Q.18 200 rad/sec Q.19 q = Q0sin LC t + 2
Q.22
20 ≅ 2H π2
Q.23 20 V
Q.16 kMT2/(R)
Q.17
L I 2R
C πa2 R
Q.21
π/4
Q.20
Q.24 R = 100W,
2N
Q.25 C = 9.2 µF
3 π Hz
EXERCISE–II Q.1 (i) 85.22 Tm2; (ii) 56.8 V; (iii) linearly
Q.2
I(R + 2λx ) 2I 2 mλ(R + 2λx ) , + BId Bd B2d 2
Q.3 (i) 2.4 × 10−5 V (ii) from c to b Q.4 (i) 3.3 × 108 A, (ii) 1.0 × 1017 W, (iii) totally unrealistic
Q.5 21.74 V, anticlockwise
µ02 I2a 2 V Q.8 4π2 R
Q.7 φ =
Q.6
2 2 2 2 3 µ 0 I aV 3a + a n 4 = 2π 2 R
3 1 3 +n 4
Q.9
17
µ0 a +b IL ln 2π a
4B2 νd πλ
Q.10
b µ 0 hωi m N ln a 2π
Page 17 of 16 E.M.I. & A.C.
EXERCISE–I
Q.12
E R1 R1R 2 + R 2 R 3 + R 3R1
Q.13
67/32 A
Q.14 (i) i1 = i2 = 10/3 A, (ii) i1 = 50/11 A ; i2 = 30/11 A, (iii) i1 = 0, i2 = 20/11 A, (iv) i1 = i2 = 0 Q.15 C2=
ε max
C1 1 1 + ωL − ω(C1 + C 2 ) C 2
Q.17 0.08 H, 17.28 W
Q.20 0.2 mH,
; C1=
ε max
C 2 C1 1 1 + ωL − C1 C 2 ω(C1 + C 2 )
Q.16 42 + 20t volt
Q.19 77Ω, 97.6Ω, 7.7V, 9.76V
Q.18 2A, 400W
1 µF, 8 × 105 rad/s 32
EXERCISE–III Q.1 IEA=
7 3 1 A ; IBE= A ; IFE = A 22 11 22
Q.3 V = 1 ms−1, R1 = 0.47 Ω, R2 = 0.30 Ω
Q.2 A, D
Q.4 ACD
[
]
Bωr 2 1−e − Rt / L mgr 1 ωB2 r 4 2 , (ii) τ = cos ωt + (1 − e−Rt/L) Q.5 (a) E = Bωr (b) (i) I = 2R 2 2 4R
Q.6 t =
L ln 2 = 3.47 sec R
mgR
Q.9 (i) Vterminal =
2
B Z
2
; (ii)
Q.8 evB kˆ , ABDC
Q.7 D
g 2
Q.10 1.6 π × 10–13 A = 50.3 pA
Q.11 (i) A, B, C, D, (ii) B, (iii) B, (iv) (a)104A/s (b) 0 (c) 2A (d) 100 3 µC
Q.12 C Q.15 (a) i =
Q.13 D
Q.14 D
B0av in anticlockwise direction, v = velocity at time t, (b) Fnett=B02a2V/R, R
B2a 2t − 0 mgR (c) V = 2 2 1 − e mR B0 a
18
Page 18 of 16 E.M.I. & A.C.
l dB l2 R2 − 2 dt 4
Q.11
Q.17 12e–5t, 6e–10t
Q.18 D
Q.19 B
Q.20 (a) Imax =
µ 0a
π
CI 0ω2ln 2 , (b)
Page 19 of 16 E.M.I. & A.C.
Q.16 A
Q.21 B
V, I
v = 220 2 sin ωt i = 20 sin (ωt-π/4)
Q.22 20 A,
π 1 , ∴ Steady state current i = 20sin π100t − 4 4
Q.23 A
Q.24 I =
Q.25 A,C
Q.26 B
Q.27 D
Q.29 D
Q.30 D
Q.31 C
20 O -10 2
T T/8 T/4
T/2 5T/8
(µ 0 ni 0ω cos ωt )πa 2 (Ld) ρ2πR
Q.28 C
Q.32 (A) P; (B) P, Q, S; (C) Q,S ; (D) Q, R, S
19
9T/8 t
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: PHYSICS TOPIC: XII P7. Optics Index: 1. Key Concepts 2. Exercise I 3. Exercise II 4. Exercise III 5. Exercise IV 6. Answer Key 7. 34 Yrs. Que. from IIT-JEE 8. 10 Yrs. Que. from AIEEE
1
1. (i)
LAWS OF REFLECTION : The incident ray (AB), the reflected ray (BC) and normal (NN') to the surface (SC') of reflection at the point of incidence (B) lie in the same plane. This plane is called the plane of incidence (also plane of reflection).
(ii)
The angle of incidence (the angle between normal and the incident ray) and the angle of reflection (the angle between the reflected ray and the normal) are equal ∠i = ∠r
2. (a) (b)
OBJECT : Real : Point from which rays actually diverge. Virtual : Point towards which rays appear to converge
3.
IMAGE : Image is decided by reflected or refracted rays only. The point image for a mirror is that point Towards which the rays reflected from the mirror, actually converge (real image). OR From which the reflected rays appear to diverge (virtual image) .
(i) (ii) 4. (a) (b) (c) 5.
CHARACTERISTICS OF REFLECTION BY A PLANE MIRROR : The size of the image is the same as that of the object. For a real object the image is virtual and for a virtual object the image is real. For a fixed incident light ray, if the mirror be rotated through an angle θ the reflected ray turns through an angle 2θ. SPHERICAL MIRRORS :
6.
Concave Convex PARAXIAL RAYS : Rays which forms very small angle with axis are called paraxial rays.
7.
SIGN CONVENTION : We follow cartesian co-ordinate system convention according to which (a) The pole of the mirror is the origin . (b) The direction of the incident rays is considered as positive x-axis. (c) Vertically up is positive y-axis. Note : According to above convention radius of curvature and focus of concave mirror is negative and of convex mirror is positive. 1 1 1 8. MIRROR FORMULA : = + . f v u f = x- coordinate of focus ; u = x-coordinate of object ; v = x-coordinate of image Note : Valid only for paraxial rays.
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KEY CONCEPTS
TRANSVERSE MAGNIFICATION : m =
h2 =−v h1 u
h2 = y co-ordinate of images h1 = y co-ordinate of the object (both perpendicular to the principle axis of mirror) 10.
NEWTON'S FORMULA : Applicable to a pair of real object and real image position only . They are called conjugate positions or foci. X,Y are the distance along the principal axis of the real object and real image respectively from the principal focus . XY = f 2
11.
OPTICAL POWER : Optical power of a mirror (in Diopters) = –
1 ; f
f = focal length (in meters) with sign .
REFRACTION -PLANE SURFACE 1. (i) (ii)
LAWS OF REFRACTION (AT ANY REFRACTING SURFACE) : The incident ray (AB), the normal (NN') to the refracting surface (II') at the point of incidence (B) and the refracted ray (BC) all lie in the same plane called the plane of incidence or plane of refraction . Sin i = Constant : Sin r
for any two given media and for light of a given wave length. This is known as SNELL'S Law . Sin i n v λ = 1n2 = 2 = 1 = 1 Sin r λ2 n1 v2
Note : Frequency of light does not change during refraction . 2.
DEVIATION OF A RAY DUE TO REFRACTION :
3. (i)
REFRACTION THROUGH A PARALLEL SLAB : Emerged ray is parallel to the incident ray, if medium is same on both sides.
(ii)
Lateral shift x =
t sin(i − r) cos r
t = thickness of slab Note : Emerged ray will not be parallel to the incident ray if the medium on both the sides are different .
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9.
APPARENT DEPTH OF SUBMERGED OBJECT : Page 4 of 20 GEOMETRICAL OPTICS
4.
(h′ < h) at near normal incidence h′ =
µ2 h µ1
Note : h and h' are always measured from surface. 5.
(i) (ii)
CRITICAL ANGLE & TOTAL INTERNAL REFLECTION ( T. I. R.)
CONDITIONS OF T. I. R. Ray going from denser to rarer medium Angle of incidence should be greater than the critical angle (i > c) . n Critical angle C = sin-1 r ni
6.
REFRACTION THROUGH PRISM :
1. 2. 3. 4.
δ = (i + i′) - (r + r′) r + r′ = A Variation of δ versus i (shown in diagram) . There is one and only one angle of incidence for which the angle of deviation is minimum. When δ = δm then i = i′ & r = r′ , the ray passes symetrically about the prism, & then sin
n=
[
A + δm 2
sin
[ ] A 2
] , where n = absolute R.I. of glass .
Note : When the prism is dipped in a medium then n = R.I. of glass w.r.t. medium .
4
7. 8.
For a thin prism ( A ≤10o) ; δ = ( n – 1 ) A DISPERSION OF LIGHT : The angular spilitting of a ray of white light into a number of components when it is refracted in a medium other than air is called Dispersion of Light. Angle of Dispersion : Angle between the rays of the extreme colours in the refracted (dispersed) light is called Angle of Dispersion . θ = δv – δr . Dispersive power (ω) of the medium of the material of prism . ω=
angular dispersion deviation of mean ray (yellow)
For small angled prism ( A ≤10o ) ω=
n + nR n −n δv − δR = v R ;n= v δy n −1 2
nv, nR & n are R. I. of material for violet, red & yellow colours respectively . 9. (i)
COMBINATION OF TWO PRISMS : ACHROMATIC COMBINATION : It is used for deviation without dispersion . Condition for this (nv - nr) A = (n′v - n′r) A′ . nv + nR
Net mean deviation =
2
n ′v + n ′R − 1 A – − 1 A′ . 2
or ωδ + ω′δ′ = 0 where ω, ω′ are dispersive powers for the two prisms & δ , δ′ are the mean deviation. (ii)
DIRECT VISION COMBINATION : It is used for producing disperion without deviation condition n v + nR
for this
2
n ′v + n ′R − 1 A = − 1 A′ . 2
Net angle of dispersion = (nv - nr) A = (nv′ - nr′) A′ .
REFRACTION AT SPERICAL SURFACE 1.(a)
(b) 2. (a)
µ 2 µ1 µ 2 − µ1 − = v u R v, u & R are to be kept with sign as v = PI u = –PO R = PC (Note radius is with sign) µ1 v m= µ u 2 LENS FORMULA : 1 1 1 − = v u f
(b)
1 = (µ – 1) f
(c)
m=
1 1 − R1 R 2
v u
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5. 6.
Q.1
A plane mirror 50 cm long , is hung parallel to a vertical wall of a room, with its lower edge 50 cm above the ground. A man stands infront of the mirror at a distance 2 m away from the mirror. If his eyes are at a height 1.8 m above the ground, find the length of the floor between him & the mirror, visible to him reflected from the mirror.
Q.2
In figure shown AB is a plane mirror of length 40cm placed at a height 40cm from ground. There is a light source S at a point on the ground. Find the minimum and maximum height of a man (eye height) required to see the image of the source if he is standing at a point A on ground shown in figure.
Q.3
A plane mirror of circular shape with radius r = 20 cm is fixed to the ceiling. A bulb is to be placed on the axis of the mirror. A circular area of radius R = 1 m on the floor is to be illuminated after reflection of light from the mirror. The height of the room is 3m. What is maximum distance from the center of the mirror and the bulb so that the required area is illuminated?
Q.4
A light ray I is incident on a plane mirror M. The mirror is rotated in the 9 direction as shown in the figure by an arrow at frequency rev/sec. π The light reflected by the mirror is received on the wall W at a distance 10 m from the axis of rotation. When the angle of incidence becomes 37° find the speed of the spot (a point) on the wall?
Q.5
A concave mirror of focal length 20 cm is cut into two parts from the middle and the two parts are moved perpendicularly by a distance 1 cm from the previous principal axis AB. Find the distance between the images formed by the two parts?
Q.6
A balloon is rising up along the axis of a concave mirror of radius of curvature 20 m. A ball is dropped from the balloon at a height 15 m from the mirror when the balloon has velocity 20 m/s. Find the speed of the image of the ball formed by concave mirror after 4 seconds? [Take : g=10 m/s2]
Q.7
A thin rod of length d/3 is placed along the principal axis of a concave mirror of focal length = d such that its image, which is real and elongated, just touches the rod. Find the length of the image?
Q.8
A point object is placed 33 cm from a convex mirror of curvature radius = 40 cm. A glass plate of thickness 6 cm and index 2.0 is placed between the object and mirror, close to the mirror. Find the distance of final image from the object?
Q.9
A long solid cylindrical glass rod of refractive index 3/2 is immersed in a 3 3 . The ends of the rod are perpendicular 4 to the central axis of the rod. a light enters one end of the rod at the central axis as shown in the figure. Find the maximum value of angle θ for which internal reflection occurs inside the rod?
liquid of refractive index
6
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EXERCISE # I
Q.11
A ray of light enters a diamond (n = 2) from air and is being internally reflected near the bottom as shown in the figure. Find maximum value of angle θ possible?
Q.12 A ray of light falls on a transparent sphere with centre at C as shown in figure. The ray emerges from the sphere parallel to line AB. Find the refractive index of the sphere. Q.13 A beam of parallel rays of width b propagates in glass at an angle θ to its plane face . The beam width after it goes over to air through this face is _______ if the refractive index of glass is µ.
Q.14 A cubical tank (of edge l) and position of an observer are shown in the figure. When the tank is empty, edge of the bottom surface of the tank is just visible. An insect is at the centre C of its bottom surface. To what height a transparent liquid of refractive index µ =
5 must be poured in the tank so that the insect will 2
become visible? Q.15 Light from a luminous point on the lower face of a 2 cm thick glass slab, strikes the upper face and the totally reflected rays outline a circle of radius 3.2 cm on the lower face. What is the refractive index of the glass. Q.16 A ray is incident on a glass sphere as shown. The opposite surface of the sphere is partially silvered. If the net deviation of the ray transmitted at the partially silvered surface is 1/3rd of the net deviation suffered by the ray reflected at the partially silvered surface (after emerging out of the sphere). Find the refractive index of the sphere. Q.17 A narrow parallel beam of light is incident on a transparent sphere of refractive index 'n'. If the beam finally gets focussed at a point situated at a distance = 2 × (radius of sphere) from the centre of the sphere, then find n? Q.18 A uniform, horizontal beam of light is incident upon a quarter cylinder of radius R = 5 cm, and has a refractive index 2 3 . A patch on the table for a distance 'x' from the cylinder is unilluminated. find the value of 'x'?
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Q.10 A slab of glass of thickness 6 cm and index 1.5 is place somewhere in between a concave mirror and a point object, perpendicular to the mirror's optical axis. The radius of curvature of the mirror is 40 cm. If the reflected final image coincides with the object, then find the distance of the object from the mirror?
Q.20 An object is kept at a distance of 16 cm from a thin lens and the image formed is real. If the object is kept at a distance of 6 cm from the same lens the image formed is virtual. If the size of the image formed are equal, then find the focal length of the lens? Q.21 A thin convex lens forms a real image of a certain object ‘p’ times its size. The size of real image becomes ‘q’ times that of object when the lens is moved nearer to the object by a distance ‘a’ find focal length of the lens ? Q.22 In the figure shown, the focal length of the two thin convex lenses is the same = f. They are separated by a horizontal distance 3f and their optical axes are displaced by a vertical separation 'd' (d n1) . (ii)
∆E = (13.6 ev) 1 2 − 1 2 . n 1
∆E = hν
;
n 2
ν = frequency of spectral line emitted .
1 = ν = wave no. [ no. of waves in unit length (1m)] = R 1 2 − 1 2 λ n2 n 1
.
Where R = Rydberg's constant for hydrogen = 1.097 × 107 m-1 . (v)
For hydrogen like atom/spicies of atomic number Z :
n2 Z2 ; Enz = (– 13.6) 2 ev Z n Rz = RZ2 – Rydberg's constant for element of atomic no. Z .
rnz =
Bohr radius Z
n2 = (0.529 Aº)
Note : If motion of the nucleus is also considered , then m is replaced by µ .
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6. (a)
Where µ = reduced mass of electron - nucleus system = mM/(m+M) .
1 1 ν=R 2 − n 22 1
Ultraviolet region (ii)
1 1 − 2 n 22 2
;
n2 > 2
;
n2 > 3
1 1 − 2 ; 2 n 2 4
n2 > 4
1 1 − 2 n 22 3
Bracket Series : (Landing orbit n = 4)
In the mid infrared region ν = R (v)
Pfund Series : (Landing orbit n = 5) In far infrared region ν = R 12 − 1 2 5
In all these series n2
8.
n2 > 1
Paschan Series : (Landing orbit n = 3)
In the near infrared region ν = R (iv)
;
Balmer Series : (Landing orbit n = 2)
Visible region ν = R (iii)
Page 4 of 20 MORDERN PHYSICS
7. (i)
2 In this case En = (–13.6 ev) Z . µ n 2 me SPECTRAL SERIES : Lyman Series : (Landing orbit n = 1) .
;
n 2
= n1 + 1 is the α line = n1 + 2 is the β line = n1 + 3 is the γ line ........... etc .
n2 > 5
where n1 = Landing orbit
EXCITATION POTENTIAL OF ATOM :
Excitation potential for quantum jump from n1 → n2 =
E n 2 −E n1 electronch arg e
.
9.
IONIZATION ENERGY : The energy required to remove an electron from an atom . The energy required to ionize hydrogen atom is = 0 - ( - 13.6) = 13.6 ev .
10.
IONIZATION POTENTIAL :
Potential difference through which an electron is moved to gain ionization energy = 11. (i) (ii) (iii) (iv)
X - RAYS : Short wavelength (0.1 Aº to 1 Aº) electromagnetic radiation . Are produced when a metal anode is bombarded by very high energy electrons . Are not affected by electric and magnetic field . They cause photoelectric emission . Characteristics equation eV = hνm e = electron charge ; V = accelerating potential νm = maximum frequency of X - radiation
4
−E n
electronicch arg e
(vii) (viii)
Intensity of X - rays depends on number of electrons hitting the target . Cut off wavelength or minimum wavelength, where v (in volts) is the p.d. applied to the tube 12400 Aº . λ min ≅ V Continuous spectrum due to retardation of electrons . Characteristic Spectrum due to transition of electron from higher to lower ν α (z - b)2 ; υ = a (z - b)2 [ MOSELEY'S LAW ] ; b = 7.4 for L series b = 1 for K series Where b is Shielding factor (different for different series) .
Note : (i)
12.
Binding energy = - [ Total Mechanical Energy ] c 137 n
(ii)
Vel. of electron in nth orbit for hydrogen atom ≅
(iii)
For x - rays
(iv)
Series limit of series means minimum wave length of that series.
;
c = speed of light .
1 1 1 =R(z−b)2 2 − 2 λ n1 n 2
NUCLEAR DIMENSIONS : R = Ro A1/3 Where Ro = empirical constant = 1.1 × 10−15 m ;
A = Mass number of the atom
13.
RADIOACTIVITY : The phenomenon of self emission of radiation is called radioactivity and the substances which emit these radiations are called radioactive substances . It can be natural or artificial (induced) .
14. (i)
α , β , γ RADIATION : α − particle : (a) Helium nucleus (2He4) (c) Velocity 106 − 107 m/s
(ii)
β − particle : (a) Have much less energy ;
(b) energy varies from 4 Mev to 9 Mev ; (d) low penetration
; ;
(b) more penetration ; (c) higher velocities than α particles
(iii)
γ − radiation : Electromagnetic waves of very high energy .
15. (A)
LAWS OF RADIOACTIVE DISINTEGRATION : DISPLACEMENT LAW : In all radioactive transformation either an α or β particle (never both or more than one of each simultaneously) is emitted by the nucleus of the atom.
(B)
A− 4 z − 2Y
(i)
α − emission : zXA →
+ 2α4 + Energy
(ii)
β − emission : zXA → β +
(iii)
γ − emission : emission does not affect either the charge number or the mass number .
A z + 1Y
+ ν (antinuetrino)
STASTISTICAL LAW : The disintegration is a random phenomenon . Whcih atom disintegrates first is purely a matter of chance . Number of nuclei disintegrating per second is given ; (disintegration /s /gm is called specific activity) . (i)
dN dN αN → =−λN = activity . dt dt
Where N = No. of nuclei present at time t (ii)
N = No
e− λ t
;
λ = decay constant
No = number of nuclei present in the beginning .
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Page 5 of 20 MORDERN PHYSICS
(v) (vi)
Half life of the population T1/2 =
0.693 λ
;
at the end of n half−life periods the number of nuclei left N =
16.
18.
1 λ
(iv)
MEAN
(v)
CURIE : The unit of activity of any radioactive substance in which the number of disintegration per second is 3.7 ×1010 .
LIFE OF AN ATOM
=
; Tav =
ATOMIC MASS UNIT ( a.m.u. OR U) :
1 amu = 17.
Σlifetimeof allatoms total number of atoms
No . 2n
1 × (mass of carbon − 12 atom) = 1.6603 × 10−27 kg 12
MASS AND ENERGY : The mass m of a particle is equivalent to an energy given by E = mc2 ; c = speed of light . 1 amu = 931 Mev MASS DEFECT AND BINDING ENERGY OF A NUCLEUS : The nucleus is less massive than its constituents . The difference of masses is called mass defect . ∆ M = mass defect = [ Zmp + (A − Z) mn] − MzA .
Total energy required to be given to the nucleus to tear apart the individual nucleons co mp o s in g the nucleus , away from each other and beyond the range of interaction forces is called the Binding Energy of a nucleus . B.E. = (∆ M)C2 . B.E. per nucleon =
( ∆ M) C 2
. A Greater the B.E. , greater is the stability of the nucleus .
19. (i) (ii) (iii)
NUCLEAR FISSION : Heavy nuclei of A , above 200 , break up onto two or more fragments of comparable masses. The total B.E. increases and excess energy is released . The man point of the fission energy is leberated in the form of the K.E. of the fission fragments
. eg. 20. (i) (ii) (iii)
92 1 235 1 236 141 92 U + o n → 92 U→ 56 Ba + 36 Kr +3 o n
+ energy
NUCLEAR FUSION ( Thermo nuclear reaction) : Light nuclei of A below 20 , fuse together , the B.E. per nucleon increases and hence the excess energy is released . These reactions take place at ultra high temperature ( ≅ 107 to 109) Energy released exceeds the energy liberated in the fission of heavy nuclei .
eg . 411P→14 He+ 0+1e . (Positron) (iv)
The energy released in fusion is specified by specifying Q value . i.e. Q value of reaction = energy released in a reaction .
Note : (i) (ii)
In emission of β− , z increases by 1 . In emission of β+ , z decreases by 1 .
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Page 6 of 20 MORDERN PHYSICS
(iii)
Q.1
A parallel beam of uniform, monochromatic light of wavelength 2640 A has an intensity of 200W/m2. The number of photons in 1mm3 of this radiation are ........................
Q.2
When photons of energy 4.25eV strike the surface of a metal A, the ejected photoelectrons have maximum kinetic energy Ta eV and de Broglie wavelength λa. The maximum kinetic energy of photoelectrons liberated from another metal B by photons of energy 4.7eV is Tb = (Ta – 1.5) eV. If the De Broglie wavelength of these photoelectrons is λb = 2λa, then find The work function of a (b) The work function of b is (c) Ta and Tb
(a) Q.3
(a)
When a monochromatic point source of light is at a distance of 0.2 m from a photoelectric cell, the cut off voltage and the saturation current are respectively 0.6 volt and 18.0 mA. If the same source is placed 0.6 m away from the photoelectric cell, then find the stopping potential (b) the saturation current
Q.4
An isolated metal body is illuminated with monochromatic light and is observed to become charged to a steady positive potential 1.0 V with respect to the surrounding. The work function of the metal is 3.0 eV. The frequency of the incident light is ______________.
Q.5
663 mW of light from a 540 nm source is incident on the surface of a metal. If only 1 of each 5 × 109 incident photons is absorbed and causes an electron to be ejected from the surface, the total photocurrent in the circuit is ________.
Q.6
Light of wavelength 330 nm falling on a piece of metal ejects electrons with sufficient energy which requires voltage V0 to prevent a collector. In the same setup, light of wavelength 220 nm, ejects electrons which require twice the voltage V0 to stop them in reaching a collector. Find the numerical value of voltage V0.(Take plank's constant, h = 6.6 × 10–34 Js and 1 eV = 1.6 × 10–19 J)
Q.7
A hydrogen atom in a state having a binding energy 0.85eV makes a transition to a state of excitation energy 10.2eV. The wave length of emitted photon is ....................nm.
Q.8
A hydrogen atom is in 5th excited state. When the electron jumps to ground state the velocity of recoiling hydrogen atom is ................ m/s and the energy of the photon is ............eV.
Q.9
The ratio of series limit wavlength of Balmer series to wavelength of first line of paschen series is .............
Q.10 An electron joins a helium nucleus to form a He+ ion. The wavelength of the photon emitted in this process if the electron is assumed to have had no kinetic energy when it combines with nucleus is .........nm. Q.11
Three energy levels of an atom are shown in the figure. The wavelength corresponding to three possible transition are λ1, λ2 and λ3. The value of λ3 in terms of λ1 and λ2 is given by ______.
Q.12 Imagine an atom made up of a proton and a hypothetical particle of double the mass of an electron but having the same charge as the electron. Apply the Bohr atom model and consider a possible transitions of this hypothetical particle to the first excited level. Find the longest wavelngth photon that will be emitted λ (in terms of the Rydberg constant R.) Q.13 In a hydrogen atom, the electron moves in an orbit of radius 0.5 Å making 1016 revolution per second. The magnetic moment associated with the orbital motion of the electron is _______. Q.14 The positron is a fundamental particle with the same mass as that of the electron and with a charge equal to that of an electron but of opposite sign. When a positron and an electron collide, they may annihilate each other. The energy corresponding to their mass appears in two photons of equal energy. Find the wavelength of the radiation emitted. [Take : mass of electron = (0.5/C2)MeV and hC = 1.2×10–12 MeV.m where h is the Plank's constant and C is the velocity of light in air]
7
Page 7 of 20 MORDERN PHYSICS
EXERCISE # I
Q.16
The surface of cesium is illuminated with monochromatic light of various wavelengths and the stopping potentials for the wavelengths are measured. The results of this experiment is plotted as shown in the figure. Estimate the value of work function of the cesium and Planck’s constant.
Q.17 A hydrogen like atom has its single electron orbiting around its stationary nucleus. The energy to excite the electron from the second Bohr orbit to the third Bohr orbit is 47.2 eV. The atomic number of this nucleus is ______________. Q.18 A single electron orbits a stationary nucleus of charge Ze where Z is a constant and e is the electronic charge. It requires 47.2eV to excite the electron from the 2nd Bohr orbit to 3rd Bohr orbit. Find (i) the value of Z, (ii) energy required to excite the electron from the third to the fourth orbit (iii) the wavelength of radiation required to remove the electron from the first orbit to infinity (iv) the kinetic energy, potential energy and angular momentum in the first Bohr orbit (v) the radius of the first Bohr orbit. Q.19 A hydrogen like atom (atomic number Z) is in higher excited state of quantum number n. This excited atom can make a transition to the first excited state by successively emitting two photons of energy 22.95eV and 5.15eV respectively. Alternatively, the atom from the same excited state can make transition to the second excited state by successively emitting two photons of energies 2.4eV and 8.7eV respectively. Find the values of n and Z. Q.20 Find the binding energy of an electron in the ground state of a hydrogen like atom in whose spectrum the third of the corresponding Balmer series is equal to 108.5nm. Q.21 Which level of the doubly ionized lithium has the same energy as the ground state energy of the hydrogen atom. Find the ratio of the two radii of corresponding orbits. Q.22 The binding energies per nucleon for deuteron (1H2) and helium (2He4) are 1.1 MeV and 7.0 MeV respectively. The energy released when two deuterons fuse to form a helium nucleus (2He4) is ________. Q.23 A radioactive decay counter is switched on at t = 0. A β - active sample is present near the counter. The counter registers the number of β - particles emitted by the sample. The counter registers 1 × 105 β - particles at t = 36 s and 1.11 × 105 β - particles at t = 108 s. Find T½ of this sample 40 40 Q.24 An isotopes of Potassium 19 Ar which is stable. K has a half life of 1.4 × 109 year and decays to Argon 18 (i) Write down the nuclear reaction representing this decay. (ii) A sample of rock taken from the moon contains both potassium and argon in the ratio 1/7. Find age of rock
Q.25 At t = 0, a sample is placed in a reactor. An unstable nuclide is produced at a constant rate R in the sample by neutron absorption. This nuclide β— decays with half life τ. Find the time required to produce 80% of the equilibrium quantity of this unstable nuclide. Q.26 Suppose that the Sun consists entirely of hydrogen atom and releases the energy by the nuclear reaction, 4 11H → 42 He with 26 MeV of energy released. If the total output power of the Sun is assumed to remain constant at 3.9 × 1026 W, find the time it will take to burn all the hydrogen. Take the mass of the Sun as 1.7 × 1030 kg.
8
Page 8 of 20 MORDERN PHYSICS
Q.15 A small 10W source of ultraviolet light of wavelength 99 nm is held at a distance 0.1 m from a metal surface. The radius of an atom of the metal is approximately 0.05 nm. Find (i) the average number of photons striking an atom per second. (ii) the number ofphotoelectrons emitted per unit area per second if the efficiency of liberation ofphotoelectrons is 1%.
C13 + 1H1 → 7N14
6
N14 + 1H1 → 8O15 → 7N15 + +1e0
7
N15 + 1H1 → 6C12 + 2He4 Find how many tons of hydrogen must be converted every second into helium . The solar constant is 8 J / cm2 min. Assume that hydrogen forms 35% of the sun's mass . Calculate in how many years this hydrogen will be used up if the radiation of the sun is constant . me = 5.49 × 10-4 amu, atomic masses mH=1.00814 amu, mHe=4.00388 amu, mass of the sun=2 × 1030 kg, distance between the sun and the earth= 1.5× 1011m. 1 amu = 931 MeV. 7
Q.28 An electron of mass "m" and charge "e" initially at rest gets accelerated by a constant electric field E. The rate of change of DeBroglie wavelength of this electron at time t is ................. List of recommended questions from I.E. Irodov. 5.247, 5.249, 5.260, 5.262, 5.263, 5.264, 5.265, 5.266, 5.270, 5.273, 5.277 6.21, 6.22, 6.27, 6.28, 6.30, 6.31, 6.32, 6.33, 6.35, 6.37, 6.38, 6.39, 6.40, 6.41, 6.42, 6.43, 6.49, 6.50, 6.51, 6.52, 6.53, 6.133, 6.134, 6.135, 6.136, 6.137, 6.138, 6.141, 6.214, 6.233, 6.249, 6.264, 6.289
EXERCISE # II Q.1 (a) (b)
Find the force exerted by a light beam of intensity I, incident on a cylinder (height h and base radius R) placed on a smooth surface as shown in figure if: surface of cylinder is perfectly reflecting surface of cylinder is having reflection coefficient 0.8. (assume no transmission)
Q.2
A small plate of a metal (work function = 1.17 eV) is placed at a distance of 2m from a monochromatic light source of wave length 4.8 × 10-7 m and power 1.0 watt. The light falls normally on the plate. Find the number of photons striking the metal plate per square meter per sec. If a constant uniform magnetic field of strength 10–4 tesla is applied parallel to the metal surface. Find the radius of the largest circular path followed by the emitted photoelectrons.
Q.3
Electrons in hydrogen like atoms (Z = 3) make transitions from the fifth to the fourth orbit & from the fourth to the third orbit. The resulting radiations are incident normally on a metal plate & eject photo electrons. The stopping potential for the photoelectrons ejected by the shorter wavelength is 3.95 volts. Calculate the work function of the metal, & the stopping potential for the photoelectrons ejected by the longer wavelength. (Rydberg constant = 1.094 × 107 m–1)
Q.4
A beam of light has three wavelengths 4144Å, 4972Å & 6216 Å with a total intensity of 3.6×10–3 W.m–2 equally distributed amongst the three wavelengths. The beam falls normally on an area 1.0 cm2 of a clean metallic surface of work function 2.3 eV. Assume that there is no loss of light by reflection and that each energetically capable photon ejects one electron. Calculate the number of photoelectrons liberated in two seconds.
9
Page 9 of 20 MORDERN PHYSICS
Q.27 Assuming that the source of the energy of solar radiation is the energy of the formation of helium from hydrogen according to the following cyclic reaction : C12 + 1H1 → 7N13 → 6C13 + +1e0 6
(i) Q.6
(i) (ii) (iii)
Q.7 (i) (ii)
Monochromatic radiation of wavelength λ1 = 3000Å falls on a photocell operating in saturating mode. The corresponding spectral sensitivity of photocell is J = 4.8 × 10–3 A/w. When another monochromatic radiation of wavelength λ2 = 1650Å and power P = 5 × 10–3 W is incident, it is found that maximum velocity of photoelectrons increases n = 2 times. Assuming efficiency of photoelectron generation per incident photon to be same for both the cases, calculate threshold wavelength for the cell. (ii) saturation current in second case. A monochromatic point source S radiating wavelength 6000 Å with power 2 watt, an aperture A of diameter 0.1 m & a large screen SC are placed as shown in figure . A photoemissive detector D of surface area 0.5 cm2 is placed at the centre of the screen. The efficiency of the detector for the photoelectron generation per incident photon is 0.9. Calculate the photon flux density at the centre of the screen and the photocurrent in the detector . If a concave lens L of focal length 0.6 m is inserted in the aperture as shown, find the new values of photon flux density & photocurrent Assume a uniform average transmission of 80% for the lens . If the work-function of the photoemissive surface is 1 eV, calculate the values of the stopping potential in the two cases (without & with the lens in the aperture). A small 10 W source of ultraviolet light of wavelength 99 nm is held at a distance 0.1 m from a metal surface. The radius of an atom of the metal is approximaterly 0.05 nm. Find : the number of photons striking an atom per second. the number of photoelectrons emitted per second if the efficiency of liberation of photoelectrons is 1%.
Q.8
A neutron with kinetic energy 25 eV strikes a stationary deuteron. Find the de Broglie wavelengths of both particles in the frame of their centre of mass.
Q.9
Two identical nonrelativistic particles move at right angles to each other, possessing De Broglie wavelengths, λ1 & λ2 . Find the De Broglie wavelength of each particle in the frame of their centre of mass.
Q.10 A stationary He+ ion emitted a photon corresponding to the first line its Lyman series. That photon liberated a photoelectron from a stationary hydrogen atom in the ground state. Find the velocity of the photoelectron. Q.11
(i) (ii) (iii) Q.12
A gas of identical hydrogen like atoms has some atoms in the lowest (ground) energy level A & some atoms in a particular upper (excited) energy level B & there are no atoms in any other energy level. The atoms of the gas make transition to a higher energy level by the absorbing monochromatic light of photon energy 2.7eV. Subsequently, the atoms emit radiation of only six different photon energies. Some of the emitted photons have energy 2.7 eV. Some have energy more and some have less than 2.7 eV. Find the principal quantum number of the initially excited level B. Find the ionisation energy for the gas atoms. Find the maximum and the minimum energies of the emitted photons. A hydrogen atom in ground state absorbs a photon of ultraviolet radiation of wavelength 50 nm. Assuming that the entire photon energy is taken up by the electron, with what kinetic energy will the electron be ejected ?
Q.13 A monochromatic light source of frequency ν illuminates a metallic surface and ejects photoelectrons. The photoelectrons having maximum energy are just able to ionize the hydrogen atoms in ground state. When the whole experiment is repeated with an incident radiation of frequency (5/6)ν , the photoelectrons so emitted are able to excite the hydrogen atom beam which then emits a radiation of wavelength of 1215 Å . Find the work function of the metal and the frequency ν.
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Q.5
Q.15 A classical model for the hydrogen atom consists of a single electron of mass me in circular motion of radius r around the nucleus (proton). Since the electron is accelerated, the atom continuously radiates
(i) (ii) (iii)
4 electromagnetic waves. The total power P radiated by the atom is given by P = P0 r where e6 (C = velocity of light) P0 = 96 π3ε 0 3C 3m e 2 Find the total energy of the atom. Calculate an expression for the radius r (t) as a function of time. Assume that at t = 0, the radius is r0 = 10–10 m. Hence or otherwise find the time t0 when the atom collapses in a classical model of the hydrogen atom. 2 e2 1 −15 · = r ≈ 3 × 10 m Take : e 2 3 4 π ε 0 m e C
Q.16 Simplified picture of electron energy levels in a certain atom is shown in the figure. The atom is bombarded with high energy electrons. The impact of one of these electron has caused the complete removal of K-level is filled by an electron from the L-level with a certain amount of energy being released during the transition. This energy may appear as X-ray or may all be used to eject an M-level electron from the atom. Find : (i) the minimum potential difference through which electron may be accelerated from rest to cause the ejectrion of K-level electron from the atom. (ii) energy released when L-level electron moves to fill the vacancy in the K-level. (iii) wavelength of the X-ray emitted. (iv) K.E. of the electron emitted from the M-level. Q.17 U238 and U235 occur in nature in an atomic ratio 140 : 1. Assuming that at the time of earth’s formation the two isotopes were present in equal amounts. Calculate the age of the earth. (Half life of u238 = 4.5 × 109 yrs & that of U235 = 7.13 × 108 yrs) Q.18 The kinetic energy of an α − particle which flies out of the nucleus of a Ra226 atom in radioactive disintegration is 4.78 MeV. Find the total energy evolved during the escape of the α − particle. Q.19
A small bottle contains powdered beryllium Be & gaseous radon which is used as a source of α−particles. Neutrons are produced when α−particles of the radon react with beryllium. The yield of this reaction is (1/ 4000) i.e. only one α−particle out of 4000 induces the reaction. Find the amount of radon (Rn222) originally introduced into the source, if it produces 1.2 × 106 neutrons per second after 7.6 days. [T1/2 of Rn = 3.8 days]
Q.20 An experiment is done to determine the half − life of radioactive substance that emits one β−particle for each decay process. Measurement show that an average of 8.4 β are emitted each second by 2.5 mg of the substance. The atomic weight of the substance is 230. Find the half life of the substance. Q.21 When thermal neutrons (negligible kinetic energy) are used to induce the reaction ; 10 5B
+ 10 n → 37 Li + 42 He . α − particles are emitted with an energy of 1.83 MeV.. Given the masses of boron neutron & He4 as 10.01167, 1.00894 & 4.00386 u respectively. What is the mass of 37 Li ? Assume that particles are free to move after the collision.
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Q.14 An energy of 68.0 eV is required to excite a hydrogen like atom from its second Bohr orbit to the third. The nuclear charge Ze. Find the value of Z, the kinetic energy of the electron in the first Bohr orbit and the wavelength of the electro magnetic radiation required to eject the electron from the first Bohr orbit to infinity.
(i)
Two deuterium
( D) nuclei fuse to form a tritium ( T )nucleus with a proton as product. The reaction 2 1
3 1
may be represented as D (D, p) T. (ii) (a) (b) (c)
( )
A tritium nucleus fuses with another deuterium nucleus to form a helium 42 He nucleus with neutron as another product. The reaction is represented as T(D , n) α. Find : The energy release in each stage . The energy release in the combined reaction per deuterium & What % of the mass of the initial deuterium is released in the form of energy. Given :
( D) = 2.014102 u ( P )= 1.00785 u 2 1
1 1
; ;
( T) = 3.016049 u ; ( n )= 1.008665 u 3 1
( He)= 4.002603 u 4 2
;
1 0
Q.23 A wooden piece of great antiquity weighs 50 gm and shows C14 activity of 320 disintegrations per minute. Estimate the length of the time which has elapsed since this wood was part of living tree, assuming that living plants show a C14 activity of 12 disintegrations per minute per gm. The half life of C14 is 5730 yrs. Q.24 Show that in a nuclear reaction where the outgoing particle is scattered at an angle of 90° with the direction of the bombarding particle, the Q-value is expressed as mP m – K 1 + I Q = KP 1 + I MO MO Where, I = incoming particle, P = product nucleus, T = target nucleus, O = outgoing particle. Q.25 When Lithium is bombarded by 10 MeV deutrons, neutrons are observed to emerge at right angle to the direction of incident beam. Calculate the energy of these neutrons and energy and angle of recoil of the associated Beryllium atom. Given that : m (0n1) = 1.00893 amu ; m (3Li7) = 7.01784 amu ; m (1H2) = 2.01472 amu ; and m (4Be8) = 8.00776 amu. Q.26 A body of mass m0 is placed on a smooth horizontal surface. The mass of the body is decreasing exponentially with disintegration constant λ. Assuming that the mass is ejected backward with a relative velocity v. Initially the body was at rest. Find the velocity of body after time t. Q.27 A radionuclide with disintegration constant λ is produced in a reactor at a constant rate α nuclei per sec. During each decay energy E0 is released. 20% of this energy is utilised in increasing the temperature of water. Find the increase in temperature of m mass of water in time t. Specific heat of water is S. Assume that there is no loss of energy through water surface.
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Q.22 In a fusion reactor the reaction occurs in two stages :
Q.1 (i) (ii)
A neutron of kinetic energy 65 eV collides inelastically with a singly ionized helium atom at rest . It is scattered at an angle of 90º with respect of its original direction. Find the allowed values of the energy of the neutron & that of the atom after collision. If the atom gets de-excited subsequently by emitting radiation , find the frequencies of the emitted radiation. (Given : Mass of he atom = 4×(mass of neutron), ionization energy of H atom =13.6 eV) [JEE '93]
Q.2
A hydrogen like atom (atomic number Z) is in a higher excited state of quantum number n. This excited atom can make a transition to the first excited state by successively emitting two photons of energies 10.20 eV & 17.00 eV respectively. Alternatively , the atom from the same excited state can make a transition to the second excited state by successively emitting two photons of energies 4.25 eV & 5.95 eV respectively. Determine the values of n & Z. (Ionisation energy of hydrogen atom = 13.6 eV) [JEE’94]
Q.3
Select the correct alternative(s) : When photons of energy 4.25 eV strike the surface of a metal A, the ejected photo electrons have maximum kinetic energy TAeV and de- Broglie wave length γA. The maximum kinetic energy of photo electrons liberated from another metal B by photons of energy 4.70 eV is TB = (TA - 1.50) eV. If the de-Broglie wave length of these photo electrons is γB = 2γA, then : (A) the work function of A is 2.225 eV (B) the work function of B is 4.20 eV (C) TA = 2.00 eV (D) TB = 2.75 eV [JEE’94]
Q.4
In a photo electric effect set-up, a point source of light of power 3.2 × 10-3 W emits mono energetic photons of energy 5.0 eV. The source is located at a distance of 0.8 m from the centre of a stationary metallic sphere of work function 3.0 eV & of radius 8.0 × 10-3m . The efficiency of photo electrons emission is one for every 106 incident photons. Assume that the sphere is isolated and initially neutral, and that photo electrons are instantly swept away after emission. Calculate the number of photo electrons emitted per second. Find the ratio of the wavelength of incident light to the De - Broglie wave length of the fastest photo electrons emitted. It is observed that the photo electron emission stops at a certain time t after the light source is switched on. Why ? Evaluate the time t. [JEE’95]
(a) (b) (c) (d) Q.5
An energy of 24.6 eV is required to remove one of the electrons from a neutral helium atom. The energy (In eV) required to remove both the electrons form a neutral helium atom is : (A) 38.2 (B) 49.2 (C) 51.8 (D) 79.0 [JEE’95]
Q.6
An electron, in a hydrogen like atom , is in an excited state . It has a total energy of − 3.4 eV. Calculate: (i) The kinetic energy & (ii) The De - Broglie wave length of the electron. [JEE 96]
Q.7
An electron in the ground state of hydrogen atoms is revolving in anti-clockwise direction in a circular orbit of radius R. Obtain an expression for the orbital magnetic dipole moment of the electron. The atom is placed in a uniform magnetic induction, such that the plane normal to the electron orbit make an angle of 30º with the magnetic induction. Find the torque experienced by the orbiting electron. [JEE'96]
(i) (ii)
Q.8
A potential difference of 20 KV is applied across an x-ray tube. The minimum wave length of X - rays generated is ________ . [JEE'96]
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EXERCISE # III
(ii)
Assume that the de-Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if the distance 'd' between the atoms of the array is 2 Å. A similar standing wave is again formed if 'd' is increased to 2.5 Å but not for any intermediate value of d. Find the energy of the electrons in electron volts and the least value of d for which the standing wave of the type described above can form. [JEE' 97]
Q.10(i) The work function of a substance is 4.0 eV . The longest wavelength of light that can cause photoelectron emission from this substance is approximately : (A) 540 nm (B) 400 nm (C) 310 nm (D) 220 nm (ii)
The electron in a hydrogen atom makes a transition n1 → n2, where n1 & n2 are the principal quantum numbers of the two states . Assume the Bohr model to be valid . The time period of the electron in the initial state is eight times that in the final state . The possible values of n1 & n2 are : (A) n1 = 4, n2 = 2 (B) n1 = 8, n2 = 2 (C) n1 = 8, n2 = 1 (D) n1 = 6, n2 = 3 [JEE ’98]
Q.11
A particle of mass M at rest decays into two particles of masses m1 and m2, having non-zero velocities. The ratio of the de-Broglie wavelengths of the particles, λ1/ λ2, is (A) m1/m2 (B) m2/m1 (C) 1.0 (D) √m2/√m1 [JEE ’99]
Q.12 Photoelectrons are emitted when 400 nm radiation is incident on a surface of work function 1.9eV. These photoelectrons pass through a region containing α-particles. A maximum energy electron combines with an α-particle to form a He+ ion, emitting a single photon in this process. He+ ions thus formed are in their fourth excited state. Find the energies in eV of the photons, lying in the 2 to 4eV range, that are likely to be emitted during and after the combination. [Take , h = 4.14 × 10-15 eV−s ] [JEE ’99] Q.13(a) Imagine an atom made up of a proton and a hypothetical particle of double the mass of the electron but having the same charge as the electron. Apply the Bohr atom model and consider all possible transitions of this hypothetical particle to the first excited level. The longest wavelength photon that will be emitted has wavelength λ (given in terms of the Rydberg constant R for the hydrogen atom) equal to (A) 9/(5R) (B) 36/(5R) (C) 18/(5R) (D) 4/R [JEE’ 2000 (Scr)] (b)
The electron in a hydrogen atom makes a transition from an excited state to the ground state. Which of the following statements is true? (A) Its kinetic energy increases and its potential and total energies decrease. (B) Its kinetic energy decreases, potential energy increases and its total energy remains the same. (C) Its kinetic and total energies decrease and its potential energy increases. (D) Its kinetic, potential and total energies decrease. [JEE’ 2000 (Scr)]
Q.14(a) A hydrogen - like atom of atomic number Z is in an excited state of quantum number 2 n. It can emit a maximum energy photon of 204 eV. If it makes a transition to quantum state n, a photon of energy 40.8 eV is emitted. Find n, Z and the ground state energy (in eV) for this atom. Also, calculate the minimum energy (in eV) that can be emitted by this atom during de-excitation. Ground state energy of hydrogen atom is − 13.6 eV. [JEE' 2000] (b)
When a beam of 10.6 eV photon of intensity 2 W/m2 falls on a platinum surface of area 1 × 104 m2 and work function 5.6 ev, 0.53% of the incident photons eject photoelectrons. Find the number of photoelectrons emitted per sec and their minimum and maximum energies in eV. [JEE' 2000]
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Q.9(i) As per Bohr model, the minimum energy (in eV) required to remove an electron from the ground state of doubly ionized Li atom (Z = 3) is (A) 1.51 (B) 13.6 (C) 40.8 (D) 122.4
Q.16 A Hydrogen atom and Li++ ion are both in the second excited state. If lH and lLi are their respective electronic angular momenta, and EH and ELi their respective energies, then (A) lH > lLi and |EH| > |ELi| (B) lH = lLi and |EH| < |ELi| (C) lH = lLi and |EH| > |ELi| (D) lH < lLi and |EH| < |ELi| [JEE 2002 (Scr)] Q.17 A hydrogen like atom (described by the Bohr model) is observed to emit six wavelengths, originating from all possible transition between a group of levels. These levels have energies between – 0.85 eV and – 0.544 eV (including both these values) (a) Find the atomic number of the atom. (b) Calculate the smallest wavelength emitted in these transitions. [JEE' 2002] Q.18 Two metallic plates A and B each of area 5 × 10–4 m2, are placed at a separation of 1 cm. Plate B carries a positive charge of 33.7 × 10–12 C. A monochromatic beam of light, with photons of energy 5 eV each, starts falling on plate A at t = 0 so that 1016 photons fall on it per square meter per second. Assume that one photoelectron is emitted for every 106 incident photons. Also assume that all the emitted photoelectrons are collected by plate B and the work function of plate A remains constant at the value 2 eV. Determine (a) the number of photoelectrons emitted up to t = 10 sec. (b) the magnitude of the electric field between the plates A and B at t = 10 s and (c) the kinetic energy of the most energetic photoelectron emitted at t = 10 s when it reaches plate B. (Neglect the time taken by photoelectron to reach plate B) [JEE' 2002] Q.19 The attractive potential for an atom is given by v = v0 ln (r / r0 ) , v0 and r0 are constant and r is the radius of the orbit. The radius r of the nth Bohr's orbit depends upon principal quantum number n as : (A) r ∝ n (B) r ∝ 1/n2 (C) r ∝ n2 (D) r ∝ 1/n [JEE' 2003 (Scr)] Q.20 Frequency of a photon emitted due to transition of electron of a certain elemrnt from L to K shell is found to be 4.2 × 1018 Hz. Using Moseley's law, find the atomic number of the element, given that the [JEE' 2003] Rydberg's constant R = 1.1 × 107 m–1. Q.21 In a photoelctric experiment set up, photons of energy 5 eV falls on the cathode having work function 3 eV. (a) If the saturation current is iA = 4µA for intensity 10–5 W/m2, then plot a graph between anode potential and current. (b) Also draw a graph for intensity of incident radiation of 2 × 10–5 W/m2 ? [JEE' 2003] Q.22 A star initially has 1040 deutrons. It produces energy via, the processes 1H2 + 1H2 → 1H3 + p & 1H2 +1H3 → 2He4 + n. If the average power radiated by the star is 1016 W, the deuteron supply of the star is exhausted in a time of the order of : [JEE ’93] (A) 106 sec (B) 108 sec (C) 1012 sec (D) 1016 sec Q.23 A small quantity of solution containing 24Na radionuclide (half life 15 hours) of activity 1.0 microcurie is injected into the blood of a person. A sample of the blood of volume 1 cm3 taken after 5 hours shows an activity of 296 disintegrations per minute. Determine the total volume of blood in the body of the person. Assume that the radioactive solution mixes uniformly in the blood of the person. (1 Curie = 3.7 × 1010 disintegrations per second ) [JEE’94] Q.24(i) Fast neutrons can easily be slowed down by : (A) the use of lead shielding (B) passing them through water (C) elastic collisions with heavy nuclei (D) applying a strong electric field
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Q.15 The potential difference applied to an X - ray tube is 5 kV and the current through it is 3.2 mA. Then the number of electrons striking the target per second is [JEE' 2002 (Scr.)] (A) 2 × 1016 (B) 5 × 1016 (C) 1 × 1017 (D) 4 × 1015
Consider α − particles , β − particles & γ rays , each having an energy of 0.5 MeV . Increasing order of penetrating powers , the radiations are : [JEE’94] (A) α , β , γ (B) α , γ , β (C) β , γ , α (D) γ , β , α
Q.25 Which of the following statement(s) is (are) correct ? [JEE'94] (A) The rest mass of a stable nucleus is less than the sum of the rest masses of its separated nucleons. (B) The rest mass of a stable nucleus is greater than the sum of the rest masses of its separated nucleons. (C) In nuclear fusion, energy is released by fusion two nuclei of medium mass (approximately 100 amu). (D) In nuclear fission, energy is released by fragmentation of a very heavy nucleus. Q.26 The binding energy per nucleon of 16O is 7.97 MeV & that of required to remove a neutron from 17O is : (A) 3.52 (B) 3.64 (C) 4.23
17O is 7.75 MeV . The energy in MeV
[JEE’95] (D) 7.86
Q.27 At a given instant there are 25 % undecayed radio − active nuclei in a sample. After 10 sec the number [JEE 96] of undecayed nuclei remains to 12.5 % . Calculate : (i) mean − life of the nuclei and (ii) The time in which the number of undecayed nuclear will further reduce to 6.25 % of the reduced number. Q.28 Consider the following reaction ; 2H1 + 2H1 = 4He2 + Q . [JEE 96] Mass of the deuterium atom = 2.0141 u ; Mass of the helium atom = 4.0024 u This is a nuclear ______ reaction in which the energy Q is released is ______ MeV. Q.29(a)The maximum kinetic energy of photoelectrons emitted from a surface when photons of energy 6 eV fall on it is 4 eV. The stopping potential in Volts is : (A) 2 (B) 4 (C) 6 (D) 10 (b)
(c)
In the following, column I lists some physical quantities & the column II gives approx. energy values associated with some of them. Choose the appropriate value of energy from column II for each of the physical quantities in column I and write the corresponding letter A, B, C etc. against the number (i), (ii), (iii), etc. of the physical quantity in the answer book. In your answer, the sequence of column I should be maintained . Column I Column II (i) Energy of thermal neutrons (A) 0.025 eV (ii) Energy of X−rays (B) 0.5 eV (iii) Binding energy per nucleon (C) 3 eV (iv) Photoelectric threshold of metal (D) 20 eV (E) 10 keV (F) 8 MeV 13 seconds. Its primary decay modes are spontaneous The element Curium 248 has a mean life of 10 Cm 96 fission and α decay, the former with a probability of 8% and the latter with a probability of 92%. Each fission releases 200 MeV of energy . The masses involved in α decay are as follows : 248 244 4 96 Cm = 248 .072220 u , 94 Pu = 244 .064100 u & 2 He = 4 .002603 u . Calculate the power output from a sample of 1020 Cm atoms. (l u = 931 MeV/c2) [JEE'97]
Q.30 Select the correct alternative(s) .
[JEE '98] 20 10
(i)
Let mp be the mass of a proton, mn the mass of a neutron, M1 the mass of a Ne nucleus & M2 the mass of a 40 20Ca nucleus. Then : (A) M2 = 2 M1 (B) M2 > 2 M1 (C) M2 < 2 M1 (D) M1 < 10 (mn + mp)
(ii)
The half − life of 131I is 8 days. Given a sample of 131I at time t = 0, we can assert that : (B) no nucleus will decay before t = 8 days (A) no nucleus will decay before t = 4 days (C) all nuclei will decay before t = 16 days (D) a given nucleus may decay at any time after t = 0.
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(ii)
Q.32(a) Binding energy per nucleon vs. mass number curve for nuclei is shown in the figure. W, X, Y and Z are four nuclei indicated on the curve. The process that would release energy is (A) Y → 2Z (B) W → X + Z (C) W → 2Y (D) X → Y + Z (b)
Order of magnitude of density of Uranium nucleus is, [mP = 1.67 × 10−27 kg] (A) 1020 kg/m3 (B) 1017kg/m3 (C) 1014kg/m3 (D) 1011kg/m3
(c)
22Ne nucleus, after absorbing energy, decays into two α−particles and an unknown nucleus. The unknown
nucleus is (A) nitrogen
(B) carbon
(C) boron
(D) oxygen
(d)
Which of the following is a correct statement? (A) Beta rays are same as cathode rays (B) Gamma rays are high energy neutrons. (C) Alpha particles are singly ionized helium atoms (D) Protons and neutrons have exactly the same mass (E) None
(e)
The half−life period of a radioactive element X is same as the mean−life time of another radioactive element Y. Initially both of them have the same number of atoms. Then (A) X & Y have the same decay rate initially (B) X & Y decay at the same rate always (C) Y will decay at a faster rate than X (D) X will decay at a faster rate than Y [JEE '99]
Q.33 Two radioactive materials X1 and X2 have decay constants 10λ and λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2 will be 1/e after a time (A) 1/(10λ) (B) 1/(11λ) (C) 11/(10λ) (D) 1/(9λ) [JEE ' 2000 (Scr)] Q.34 The electron emitted in beta radiation originates from [JEE’2001(Scr)] (A) inner orbits of atoms (B) free electrons existing in nuclei (C) decay of a neutron in a nucleus (D) photon escaping from the nucleus Q.35 The half - life of 215At is 100 µs. The time taken for the radioactivity of a sample of 215At to decay to 1/16th of its initial value is [JEE 2002 (Scr)] (A) 400 µs (B) 6.3 µs (C) 40 µs (D) 300 µs Q.36 Which of the following processes represents a gamma - decay? [JEE 2002 (Scr)] A A A 1 A– 3 (B) XZ + n0 → XZ –2 + c (A) XZ + γ → XZ – 1 + a + b (C) AXZ → AXZ + f (D) AXZ + e–1 → AXZ – 1 + g Q.37 The volume and mass of a nucleus are related as (A) v ∝ m (B) v ∝ 1/m (C) v ∝ m2
[JEE 2003 (Scr)] (D) v ∝ 1/m2
Q.38 The nucleus of element X (A = 220) undergoes α-decay. If Q-value of the reaction is 5.5 MeV, then the kinetic energy of α-particle is : [JEE 2003 (Scr)] (A) 5.4 MeV (B) 10.8 MeV (C) 2.7 MeV (D) None Q.39 A radioactive sample emits n β-particles in 2 sec. In next 2 sec it emits 0.75 n β-particles, what is the mean life of the sample? [JEE 2003]
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Q.31 Nuclei of a radioactive element A are being produced at a constant rate α . The element has a decay constant λ . At time t = 0, there are N0 nuclei of the element. Calculate the number N of nuclei of A at time t . (a) (b) If α=2N0λ, calculate the number of nuclei of A after one halflife of A & also the limiting value of N as t→∞. [JEE '98]
Q.41 A photon of 10.2 eV energy collides with a hydrogen atom in ground state inelastically. After few microseconds one more photon of energy 15 eV collides with the same hydrogen atom.Then what can be detected by a suitable detector. (A) one photon of 10.2 eV and an electron of energy 1.4 eV (B) 2 photons of energy 10.2 eV (C) 2 photons of energy 3.4 eV (D) 1 photon of 3.4 eV and one electron of 1.4 eV [JEE' 2005 (Scr)] Q.42 Helium nuclie combines to form an oxygen nucleus. The binding energy per nucleon of oxygen nucleus is if m0 = 15.834 amu and mHe = 4.0026 amu (B) 0 MeV (C) 5.24 MeV (D) 4 MeV (A) 10.24 MeV [JEE' 2005 (Scr)] Q.43 The potential energy of a particle of mass m is given by
E 0 ≤ x ≤1 V( x ) = 0 0 x >1 λ1 and λ2 are the de-Broglie wavelengths of the particle, when 0 ≤ x ≤ 1 and x > 1 respectively. If the total energy of particle is 2E0, find λ1/λ2. [JEE 2005] Q.44 Highly energetic electrons are bombarded on a target of an element containing 30 neutrons. The ratio of radii of nucleus to that of helium nucleus is (14)1/3. Find (a) atomic number of the nucleus (b) the frequency of Kα line of the X-ray produced. (R = 1.1× 107 m–1 and c = 3 × 108 m/s) [JEE 2005] Q.45 Given a sample of Radium-226 having half-life of 4 days. Find the probability, a nucleus disintegrates within 2 half lives. (A) 1 (B) 1/2 (C) 3/4 (D) 1/4 [JEE 2006] Q.46 The graph between 1/λ and stopping potential (V) of three metals having work functions φ1, φ2 and φ3 in an experiment of photoelectric effect is plotted as shown in the figure. Which of the following statement(s) is/are correct? [Here λ is the wavelength of the incident ray]. (A) Ratio of work functions φ1 : φ2 : φ3 = 1 : 2 : 4 (B) Ratio of work functions φ1 : φ2 : φ3 = 4 : 2 : 1 (C) tan θ is directly proportional to hc/e, where h is Planck’s constant and c is the speed of light (D) The violet colour light can eject photoelectrons from metals 2 and 3. [JEE 2006] Q.47 In hydrogen-like atom (z = 11), nth line of Lyman series has wavelength λ equal to the de-Broglie’s wavelength of electron in the level from which it originated. What is the value of n? [JEE 2006] Q.48 Match the following Columns Column 1 (A) Nuclear fusion (B) Nuclear fission (C) β–decay (D) Exothermic nuclear reaction
[JEE 2006] Column 2 (P) Converts some matter into energy (Q) Generally occurs for nuclei with low atomic number (R) Generally occurs for nuclei with higher atomic number (S) Essentially proceeds by weak nuclear forces
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Page 18 of 20 MORDERN PHYSICS
Q.40 The wavelength of Kα X-ray of an element having atomic number z = 11 is λ . The wavelength of Kα X-ray of another element of atomic number z' is 4λ. Then z' is (B) 44 (C) 6 (D) 4 [JEE' 2005 (Scr)] (A) 11
ANSWER KEY EXERCISE # I Q.1 Q.4 Q.5 Q.9
885 Q.2 (a) 2.25eV, (b) 4.2eV, (c) 2.0 eV, 0.5 eV Q.3 (a) 0.6 volt, (b) 2.0 mA when the potential is steady, photo electric emission just stop when hυ = (3 + 1)eV = 4.0 eV 5.76 × 10–11 A Q.6 15/8 V Q.7 487.06 nm Q.8 4.26 m/s, 13.2 eV λ1λ 2 7 : 36 Q.10 22.8 nm Q.11 λ + λ Q.12 18/(5R) 1 2
Q.13 1.257 ×
10–23 Am2
×10–12
Q.16
Q.14
2.48
Q.17 5
Q.18
(i) 5, 16.5 eV, 36.4 A, 340 eV, – 680 eV,
Q.19 z = 3, n = 7
Q.20
54.4 eV
Q.23 ( T1 / 2 = 10.8 sec)
Q.24
(i)
ln 5 τ Q.25 t = ln 2
Q.26
8 3 × 1018 sec Q.27
40 19 K
m Q.15
5 10 20 , 16 80π
Q.21 n = 3, 3 : 1
2 eV, 6.53 × 10–34 J-s
h 1.06 × 10–111 m 2π Q.22 23.6 MeV
40 → 18 Ar + +1e0 + ν (ii) 4.2 × 109 years
1.14 × 1018 sec
Q.28
– h/eEt2
EXERCISE # II Q.1
38 I R h 8IhR/3C 15 C
Q.2 4.8 × 1016, 4.0 cm
Q.3 1.99 eV, 0.760 V
Q.4 1.1 × 1012 Q.5 (i) 4125Å, (ii) 13.2 µA Q.6 (i) 1.33 × 1016 photons/m2 − s ; 0.096 µÅ (ii) 2.956 × 1015 photons/m2s ; 0.0213 µA (iii) 1.06 volt Q.7 (i) 5/16 photon/sec, (ii) 5/1600 electrons/sec Q.8 λdeutron = λneutron = 8.6 pm 2λ1λ 2 Q.10 3.1 × 106 m/s Q.11 (i) 2 ; (ii) 23.04 ×10–19J ; (iii) 4 → 1 , 4 → 3 Q.9 λ = λ12 +λ 2 2 Q.12 11.24 eV Q.13 6.8 eV, 5 × 1015 Hz Q.14 489.6 eV, 25.28 Å 1/ 3
Q.15
3C re 2 t 1 e2 , (ii) r0 1 − (i) – 3 8πε 0 r r 0
, (iii) 10–10 ×
100 sec 81
Q.16 (i) 1.875 × 104 V, (ii) 2.7 × 10–15 J, (iii) 0.737 Å, (iv) 2.67 × 10–15 J Q.17 Q.20 Q.22 Q.25
6.04 × 109 yrs Q.18 4.87 MeV Q.19 3.3 × 10−6 g 10 1.7 × 10 years Q.21 7.01366 amu (a) 4 MeV , 17.6 (b) 7.2 MeV (c) 0.384 % Q.23 5196 yrs Energy of neutron = 19.768 MeV ; Energy of Beryllium = 5.0007 MeV; Angle of recoil = tan–1 (1.034) or 46°
Q.26 v = uλt
Q.27
α 0.2E 0 α t − (1 − e −λ t ) λ ∆T = mS
19
EXERCISE # III Q.1
(i) Allowed values of energy of neutron = 6.36 eV and 0.312 eV ; Allowed values of energy of He atom = 17.84 eV and 16.328 eV , (ii) 18.23 x 1015 Hz , 9.846 x 1015 Hz , 11.6 x 1015 Hz Q.3 B, C Q.4 (a) 105 s–1 ; (b) 286.18 ; (d) 111 s Q.2 n = 6, Z = 3 ehB he Q.5 D Q.6 (i) KE = 3.4 eV, (ii) λ = 6.66 Å Q.7 (i) (ii) 4πm 8πm Q.8 0.61 Å Q.9 (i) D, (ii) KE ≅ 151 eV, dleast = 0.5 Å Q.10 (i) C (ii) A, D Q.11 C Q.12 during combination = 3.365 eV; after combination = 3.88 eV (5 → 3) & 2.63 eV (4 → 3) Q.13 (a) C, (b) A Q.14 (a) n = 2, z = 4; G.S.E. − 217.6 eV; Min. energy = 10.58 eV; (b) 6.25×1019 per sec, 0, 5 eV Q.16 B Q.17 3, 4052.3 nm Q.18 5×107, 2000N/C, 23 eV Q.15 A
Q.19 A
Q.20 z = 42
Q.21
Q.22 C
Q.23 6 litre
Q.24 (i) B, (ii) A
Q.26 C
Q.27 (i) t1/2 = 10 sec. , tmeans = 14.43 s (ii) 40 seconds
Q.28 Fusion , 24
Q.29 (a) B, (b) (i) − A, (ii) − E, (iii) − F, (iv) − C, (c) ≅ 33.298 µW
Q.30 (i) C, D (ii) D
Q.31
(a) N =
Q.25 A , D
1 3 N0 , 2 N0 [α (1 − e −λt )+ λ N0 e−λt] (b) 2 λ
Q.32 (a) C ; (b) B ; (c) B ; (d) E ; (e) C
Q.33 D
Q.34 C
Q.35 A
Q.37 A
Q.38 A
Q.39 1.75n = N0(1 – e–4λ), 6.95 sec,
Q.41 A
Q.42 A
Q.43
Q.45 C
Q.46 A,C
Q.47 n = 24
Q.36 C 2 4 ln 3
Q.40 C
Q.44 ν = 1.546 × 1018 Hz
2
Q.48 (A) P, Q; (B) P, R; (C) S, P; (D) P, Q, R
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STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: PHYSICS TOPIC: XII P11. Semiconductors Electronics Index: 1. Key Concepts 2. Exercise I 3. Exercise II 4. Exercise III 5. Exercise IV 6. Answer Key 7. 34 Yrs. Que. from IIT-JEE 8. 10 Yrs. Que. from AIEEE
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Electronics The study of matter in the solid state and its physical properties has contributed a lot to modern living– particularly, the science of electronics. Solids may be crystalline or amorphous –crystalline solids have long-range order in their structure while amorphous solids do not have such order. Here we will deal with crystalline solids only. Crystalline solids & their electronic properties A crystalline solid is built around a lattice, the regular, repeating mathematical points extending throughout space. The forces responsible for the regular arrangement of atoms in a lattice are similar to those in molecular bonds – covalent and ionic. A third type crystalline bond is a metallic bond: one or more of the outermost electrons in each atom become detached from the parent atom and are free to move throughout the crystal. These electrons are known as “free electrons”, and are responsible for the conduction of electricity by metals. Band structure of solids As isolated atoms are brought together to form a solid, interactions occur between neighbouring atoms. The attractive and repulsive forces between atoms find a proper balance when the proper inter-atomic spacing is reached. As this process occurs, there are important changes in the electronic energy levels and these changes lead to the varied electrical properties of solids. An electron moving within a crystal lattice is subjected to a periodic potential due to the ionic cores present in the regular arrangement of the lattice. This is very different from the potential feld by an electron within a hydrogen atom. The energy levels (or the single atom: they are distributed in a band like structure, with gaps in between. The highest energy band wave functions are highly delocalized: an electron in one of these bands tends to be free enough to move over the entire body of the crystal. This band is known as a conduction band. The wave function belonging to lower bands are not so highly localised, they are localised to within a few neighbouring atoms of the lattice. The electrons in these bands are responsible for the formation of inter-atomic bonds: the band is referred to as the valence band and the electrons, valence electrons.’ Energy bands, that are lower in energy than the valence band, have progressively decreasing widths and have properties similar to atomic levels. Their wave functions a localized to a single atom and they are, therefore, tightly bound. Metals, insulators and semiconductors The energy gap between the lowest level of the conduction band and the highest level of the valence band is known as the band gap (Eg). The band gap (Eg) and the nature of the filling of the energy levels (according to the Pauli Exclusion principle) are cheifly respeonsible for the electrical properties of solids. When there exists a large number of electrons within the conduction band as a result of the filling process, this leads to conduction. Another reason for the existence of electrons within the conduction band is the thermal excitation of electrons from the valence band. The probability of electronic at a temperature T(in Kelvin) varies as the factor e–Eg/2kT, where E0 is the band gap and k is Boltzmann’s constant. Thus materials having very small gaps (E0 < 1 meV) behave as conductors, while those having large band gaps (Eg < 5 eV) behave as insulators at ordinary temperatures. Materials (like crystalline Si, Ge) having band gaps Eg–1eV behave as semiconductors at ordinary temperatures.
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As the temperature is raised in a semiconductor, electrons from the valence band pick up thermal excitation from atomic motion within the lattice and this leads to a transition to the conduction band, if sufficient energy is transferred to the electron. For each electron that transits to the conduction band, a valency is left within the valence band. This vacancy, referred to as a hole, helps in conduction as well. When an external electric field is applied to a semiconductor sample, the electrons within the conduction band experience a force proportional to the electric field. F = qeE, where qe is the electronic charge. The acceleration of the electron is,
qe E , m being the effective mass of the electron in the conduction band. m Due to collisions between the electron and and ionic cores within the lattice, this motion leads to an effective uniform drift velocity for electrons within the electric field. a=
The electron accelerates for a time τ, the collision time, before it loses its energy to the lattice in a collision. In this model (the Drude–Lorenz model), the average drift velocity of the electron is vd = aτ =
qe E E me
If there are ne electrons per unit volume, the current density j is given by n e q e2τ j = neqevd = E me ≡ σE (by definition), Further, vd =
q eτ E = µeE (by definition) me
where µe is the ‘mobility’ of the electron in the conduction band. Conduction occurs also in the valence band. Here, the electrons ‘hop’ from one vacancy (“hole”) to another in the electric field E, causing an electric current. This current may be thought of as due to the motion of “hole” within the valence band, the “holes” imagined to possess a positive charge equal in magnitude to that on an electron. This accounts for the fact that the “holes” move in an opposite direction to electrons within the valence band. The net current density within the semiconductor is given by: ne qe2τ e nn qe2τ n j = m + m E e n = qe(neµe + nnµn)E where nh and µh are the concentration of holes and hole mobility, respectively, within the valence band. ∴ σ = qe(neµe + nhµh)
Illustration 1: Germanium has a band gap of 0.67 eV. Calculate the value of the quantity e–Eg/kT, which is related to the probability of a transition of an electron from the valence band to the conduction band, for two temperatures at 270C and 1270C. The band gap of Ge = 0.67 eV Sol: At T = 300 K (or 273 + 27)
3
1 x 300 11600 ≈ 0.026 eV & at T = 400 K (1270C) kBT ≈ 0.0345 eV e–Eg/kT = 6.4 x 10–12 at 270C and 3.7 x 10–9 at 1270C
kBT =
Intrinsic and Extrinsic Semiconductors
For intrinsic semiconductors, the concentration of electrons within the conduction band (ne) equals that of holes within the valence band (nh) Intrinsic semiconductors are usually those which do not have any impurities within them. At absolute zero (T = 0), these semiconductors do not have any electorns in the conduction band or holes within the valence band examples are pure crystalline Si, Ge, Ga, As, In Sb, etc. Extrinsic semiconduction occurs due to the introduction of excess holes or, electrons into a semiconductor (Si for example). This is done by introducing microscopic quantities of Group V elements (P, As) as impurities into the Si – lattice. These impurities are added in very small concentrations so that they do not change the Si-lattice. Being pentavalent, there exists an excess electron (in addition to the four, which form bonds) in P0. An energy level P (or As) lies just below the conduction band of Si. The excess electron (in this donor level) of P is immediately transferred to the conduction band of Si: this results in an increase in the concentration of conduction electrons – ne. However, this also results in a reduction in the number of holes, such that, nenh = n12 This type of Si with excess electrons is known as n-type Si. Addition of small quantities of acceptor type impurities (trivalent group III elements like B) leads to an empty ‘acceptor’ level just above the filled valence band. This leads to electrons getting transferred from the valence band into this acceptor level, and thus, the introduction of holes into the valence band. The relation
nenh = n12, also holds good here.
The concentration of electrons in the conduction band gets correspondingly reduced. Such semiconductors are known as p-type semiconductors. Illustration 2: A semiconductor has an electron concentration of 0.45 x 1012/ m3 and a hole concentration of 5 x 1020 / m3. Find its conductivity. ( µ e = 0.135 m2/V-S, µ h = 0.048 m2/V-S). Sol: The conductivity, σ = e(ne µ e + nh µ h) = 1.6 x 10-19 (o.45 x 1022 x 0.135 + 5 x 1020 x 0.048) = 3.84 Ω -1-m-1. A silicon sample is made into a p-type semiconductor by doping, on an average on Indium atom Exercise 1: per 5 x 107 silicon atoms. If number density of atoms in the silicon sample is 5 x 1028 atoms/m3 then find the number density of Indium atoms in silicon per cm3.
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p-n Junction When a p-type semiconductor is joined to an n-type semiconductor (both Si, Ge) the device is known as a p-n junction.
The excess elelctrons in n-type Si diffuse into the p-type Si and fill up the holes in the adjacent p-region. A small region adjoining the junction is, therefore devoid of electrons and holes therefore has very high resistivity. This region is known as the depletion region. On the application of a forward electric field (p to positive & n to negative) the width of the depletion region is reduced and concequently, a current flows across the junction easily. When the p end is connected to a negative electrode an the n end to the positive electrode of a circuit the depletion region widens and the resistance increases tremendously due to the withdrawals of charge carries. Thus the p-n junction, almost, does not conduct in the reverse direction. Therefore, a p-n junction acts like a diode (or a rectifier).
(
)
The current vs. Voltage relation for a diode is i = Is e qeV / kT − 1
Where V is the forward p.d. applied across the diode and Is is the reverse saturation current, qe is the electronic charge (in magnitude); k, the Boltzmann constant and T, the absolute temperature. The forward current becomes significant only after V > 0.7 V (for si-diodes), in practice, and this is known as the knee voltage. The reverse saturation current (Is) also depends on temperature, through this dependence is rather weak. Is is of the order of a few µA to a few mA depending on the diode. Illustration 3 : At a temperature of 300 K, a p-n juction has a saturation current of 0.6 mA. Find the current when the voltage across the diode is 1 mV, 100 mV and -1 V. Solution:
At a temperature of 300 K, the p-n junction has a saturation current IS = 0.6 x 10-3A The current voltage relation for the diode is i = iS (eqV/kBT - 1) kBT (at 300k) ≈ 0.026 eV For V = 1 mV, i = 23 µ A, V = 100 mV, i = 27.5 mA and V = -1V, i = -0.6 mA
Rectifiers (i) Half wave rectifier A half-wave rectifier circuit consists of a diode D and the load resistance RL in series, as shown in the adjacent diagram.
If Vk is the knee-voltage of the diode ( ≈ 0.7 V for si diode) and I is the current flowing during forward bais: iRL + VR = V0 sintωt, Where the RHS represents the emf applied to the circuit. ∴i=
V0 sin ωt − VR and i > 0 RL
5
The diode is in forward bias, when sinωt >
VR V0
or, sin–1 (Vk/V0) < ωt < π – sin–1 (Vk/V0) dueing the 1st half cycle. The current i flowing in the circuit.
(ii) Full wave rectifier A full wave rectifier circuit is shown in the adjacent diagram. It consists of two diodes D1 and D2 connected to a load resistance RL. An ac–voltage Vs = V0sinωt is applied across the circuit as shown. The current through RL is just a in the case of the half-wave rectifier except that it flows during both the half-cycles.
The current through the load resistance is not a smooth dc. The maximum reverse voltage across a diode is twice the peak forward voltage.
Illustration 4:
(a) (b)
A p-n juction forms part of a rectifier circuit. A voltage waveform as shown in figure is applied to the circuit. If the diode is ideal except for a drop of 0.7 V in the forward biased condition, Plot the current through the resistor as as function of time. What is the maximum current? Calculate the average heat lost in the resistance over a single cycle.
Solution: (a) In forward bias, the potential drop across the diode is 0.7 V, and the rest of the p.d. is dropped across the resistance R (=1k Ω ) 10 − 0.7 The current (maximum) = = 9.3 mA 1000 (b) The average heat lost in the resistance over a single cycle is i2R ∆ t=(9.3 x 10-3)2 x 103 x 10-1J = 8.65 x 10-3J Exercise 2: In the full-wave rectifier circuit, the diodes D1, D2 are ideal and identical. The emf Vs = 100 sin (100 π t) volt is applied as shown (t is in sec). Calculate the voltage across the diode D1 as a function of time.
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Transistor Transistors are semiconductor devices capable of power amplification. A transistor consists of a thin central layer of one type of semiconductor sandwicthed between two relatively thick pieces of the other type. Also known as the bipolar function transistor (BJT), it can be of two types, vix., pnp or npn. The npn transistor consists of a very thin piece of p-type material sandwiched between two pieces of n-type, while the pnp transistor has a central piece of n-type. The pieces at either side are called the emitter and the collector respectively while the central part is known as the base. The base is lightly doped compared with the emitter and the collector, and is only about 3-5 µm thick.
(i) Biasing of a transistor A transistor can operate in any one of the three states, depending on the voltage across its junctions. These are the active state, the cut off state and the saturation state.
State Active Cut off Saturation
Junction Emitter Base Base collector FB RB RB RB FB FB
Where FB – Forward biased, RB – Reversed biased. The active state is the basic mode of operation. It is utilized in most amplifiers and oscillators. The cut-off and saturation states are typical of transistors operation in the switching mode. Basically, in any application using a transistor, two circuits are formed. One is the input and the other is the output circuit. Operation of an npn transistor: An increase in the forward input voltage VBE (across the emitter-base junction) brings about a fall in the height of the potential barrier at the emitter junction and an increase in the current flowing across that junction, i.e. in the emitter current IE. The electrons that make up this current are injected from the emitter into the base and diffuse through the base into the collector region, thereby boosting the collector current. Since the collector junction is reverse biased, the electrons are swept to the collector. Almost all the electrons emitted from the emitter are collected by the collector. But a small fraction of electrons recombine in the base region, which constitute the base current IB. (ii) Working of a transistor In amplification we bias B.E. junction in forward and C-B junction in reverse. Base emitter junction is forward biased hence electrons are injected by the emitter into base (n-p-n). The thickness of base
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region in very small, as a result most of the electrons diffusing in to base region cross into the collector base juction. The reverse biased CB junction sweeps off electrons as they are injected into the junction. By using Kirchhoff’s law, we can write, IE = IB + IC Where IE is emitter current, IC is collector current, IB is base current. Generally we use transistor for amplification in common emitter mode and common base mode. The collector-base, current gain is defined as β=
IC , β is very large (nearly 100) and, the collector – emitter current gain is defined as IB
α=
IC , α is very close to 1, but less than 1. IE
The parameters β and α for a transistor are decided by the constriction, the doping profile and other similar manufacturing parameters; not by the baising circuit. Since IE = IB + IC ∴
1
α
IE IB = +1 IC IC
=
1
β
+1
α 1−α In an amplifier a.c. signals are amplified. Therefore, We get β =
β ac =
∆I C ∆I B
Vo RL We get voltage gain V = β . R i BE
where RL is load resistance RBE is input resistance. Since the current gain is β , Power gain = voltage gain x current gain RL =β R BE
∆I C Thrans conductance is defined as gm = ∆V BE
(iii)
Transistor as an Amplifier In order to use a transistor as an amplifier, the emitter-base junction is forward biased (FB) and the base collector junction is reverse biased (RB). In a common-emitter (CE) amplifier, the load is connected
between the collector and the emitter through d.c. supply.
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An a.c. input singnal VS is superimposed on the bias VBE. This changes VBE by an amount ∆VBE =Vs. The output is taken between the collector and the gournd. Applying Kirchoff’s voltage law on the output loop, if VS= 0. VC = VCE + ICRL Similarly, VB = VBE VS ≠ 0 , then VB + VS = VBE + ∆VBE when The change in VBE can be related to the input resistance ri and the change in IB. VS = ∆VBE = ri ∆I B
The change in IB causes a change in IC. Thus,
β ac =
∆I C ∆I B (current gain factor)
The change in IC due to a change in IB causes a change in VCE and the voltage drop across the resistor RL because VC is fixed. Thus, ∆ VC = ∆ VCE + RL ∆ IC=0 vo = ∆ VCE = -RL ∆ IC = - β acRL ∆ IB The voltage gain AV =
vo ∆VCE β R β = = − ac L = − g m RL Where g = ac = transconductance. m ri vi ∆VBE ri
Illustration 5: In the following circuit the base current IB is 10 µ A and the collector current is 5.2mA. Can this transistor circuit be used as an amplifier? In the circuit RB = 5 Ω and RL = 1 K Ω
Solution:
We know that for a transistor is CE configuration to be used as an amplifier the BE junction must be forward biased & base collector junction must be reverse biased. In the given question we are required to just check this
loop-1
IBRB - ICRL - VCB = 0 ⇒ VCB = [(5 x 103 x 10 x 10-6) - (5.2 x 10-3 x 103)]V ⇒ (0.05 - 5.2)V ......(1) ⇒ vC < vB loop - 2 VCE + ICRL - VC = 0 ⇒ VCE = (5.5 - 5.2)V = 0.3 V ⇒ VC > VE as emitter is grounded, VC = 0.3V .......(2) from (1) and (2) VB = 5.5 V ⇒ BC junction is forward biased & hence the given is transistor would not word as an amplifier Exercise 3: In the transistor circuit shown in figure direct current gain of the transistor is 80. Assuming VBE ≈ 0, calculate (a) Base current IB (b) Potential difference between collectors and emitter terminals.
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(iv) Transistor used in an oscillator circuit The function of an oscillator circuit is to produce an alternating voltage of desired frequency without applying any external input signal. This can be achieved by feeding back a portion of the output voltage of an amplifier to its input terminal as shown in the figure.
An amplifier and an LC network are the basic part of the circuits. The amplifier is just a transistor used in common emitter mode and the LC network consists of an inductor and a capacitor. The output frequency of an oscillator is the resonating frequency of L-C network which is given as f o =
1 2π LC
From the figure, vi = vs + vf Z1 where vf = kvo = Z + Z vo 1 2
k is feedback constant which represent the fraction of output voltage which is to be feedback to input. Z1 and Z2 works as voltage divider. The voltage across the Z1 is feedback in the input of oscillator. The voltage gain of the amplifier is Av =
vo vi
vo and the overall gain is A'v = v s
Now,
vo = Avvi = Av(vs + vf)
or
vo + kvo vo = Av A'v
or
A'v =
Av 1 − kAv
By properly adjusting the feed back, it is possible to get kAv = 1 which gives A’v = ∞ , or we get an output without applying any input. The oscillator generates an ac signal. Exercise 4: In a silicon transistor, the base current is changed by 20 µA . This results in a change of 0.02 V is base to emitter voltage and a change of 2 mA in the collector current. (a) Find the input resistance ri and β ac of the transistor.. (b) If this transistor is used as an amplifier with the load resistance 5 kΩ . Find the voltage gain of the amplifier.. (v) Analysis of transistor circuit Input KVL, VBB = IBRB + VBE + IERE usually RE = 0. Therefore, VBB = IBRB + VBE
.....(1)
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For silicon transistor VBE = 0.7 V Output KVL, VCC = ICRC + VCE + IERE for RE = 0, VCC = ICRC + VCE ....(2) current relationship, IC = β IB ....(3) Where β is d.c. current gain of the transistor.. ....(4) IC = α IE Where α is the a.c. current gaing of the transistor output voltage, Vo = ICRC. input voltage, Vi = IBRB. RC Vo I C RC Therefore voltage gain AV = V = I R = β R i B B B
......(5)
Exercise 5: In the circuit shown, assume β = 60 and input resistance Rin = 1000 Ω . Find the voltage gain of the amplifier..
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